From e533fd7ffc5dabbc2b98a66d2ce05d18e9048f6e Mon Sep 17 00:00:00 2001
From: "m.gonzalez" <m.gonzalez@unizar.es>
Date: Thu, 29 Aug 2024 14:26:09 +0200
Subject: [PATCH 1/4] actualizar Mult3.tex

---
 Mult3.pdf | Bin 336299 -> 350692 bytes
 Mult3.tex |  86 ++++++++++++++++++++++++++++++++++++++++++++----------
 2 files changed, 71 insertions(+), 15 deletions(-)

diff --git a/Mult3.pdf b/Mult3.pdf
index cb073f2099d0f592be2bfcf1673581170d2176a8..f02383a3b8e4a338dbf88c8c43b72a1f03eba232 100644
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diff --git a/Mult3.tex b/Mult3.tex
index f049320..43b4554 100644
--- a/Mult3.tex
+++ b/Mult3.tex
@@ -415,7 +415,7 @@ where $\omega = dx \wedge dy$ is the standard volume form of $\CC^2$.
 \begin{proposition} Let  $\pi': X' \rightarrow \CC^2$ be the minimal resolution of $D_zF$. Then, $\pi'$ is a resolution of the meromorphic function  $\mathcal{M} : \CC^2 \rightarrow \mathbb{P}^1$ defined by $\frac{b_2^3}{b_3^2}$ too.
 \end{proposition} 
 
-There are four possibilities for the behaviour of the numbers $N^0_j, N^\infty_j$ and $N^\Delta_j$ associated with an irreducible component $E_j$ of $(\pi \circ G)^{-1} \{G=0\}$
+There are four possibilities for the behaviour of the numbers $N^0_j, N^\infty_j$ and $N^\Delta_j$ associated with an irreducible component $E_j$ of $(\pi)^{-1} \{G=0\}$
 \begin{itemize}
 \item $3 N^0_j > 2N^\infty_j = N^\Delta_j$  and it is called a \emph{zero} because  in this case $\mathcal{M}(E_j)=0$;
 \item  $2 N^\infty_j > 3N^0_j = N^\Delta_j$ and it is called a \emph{pole} because in this case  $\mathcal{M}(E_j)=\infty$;
@@ -586,7 +586,7 @@ N^\Delta_i& 25& {\bf 26}& {\bf 52}& {\bf 78}& {\bf 104}& {\bf 130}& {\bf 156}& {
 \hline
 \nu_i&2&3&5&7&9&11&13&6&11\\
 \chi()&-11&1&-1&0&0&0&-2&1&-1\\
-E_i^2& -7& -2& -2& -2& -2& -2& -1&-2&-1\\
+E_i^2& -13& -2& -4& -2& -2& -2& -1&-2&-1\\
 \hline
 \end{array}$$
 
@@ -645,7 +645,7 @@ If the strata $E^\circ_{i,j} \ne \emptyset$ and $\mathcal{M}(E_i)=1$ then either
 $$x_1= x_2, \quad y_1= y_2, \quad z = z_1 - x_2^\frac{N^0_i}{2} y_2^\frac{N^0_j}{2} + \cdots$$
  such that (\ref{quasi-trinomial}) becomes
  either
-$$z_1^3 + x_2^{\frac{N^0_i}{2}} y_2^{\frac{N^0_j}{2}} u'_1(x_2,y_2) z_1^2   + x_2^{N_i^\infty + M_i^\Delta} y_2^{N_i^\infty + M_i^\Delta} u'_2(x_2,y_2)$$
+$$z_1^3 + x_2^{\frac{N^0_i}{2}} y_2^{\frac{N^0_j}{2}} u'_1(x_2,y_2) z_1^2   + x_2^{N_i^\infty + M_i^\Delta} y_2^{N_j^\infty + M_j^\Delta} u'_2(x_2,y_2)$$
 if both  $\mathcal{M}(E_i) =  \mathcal{M}(E_j)=1$
 or
 $$z_1^3 + x_2^{\frac{N^0_i}{2}}y_2^{\frac{N^0_j}{2}} u'_1(x_2,y_2) z_1^2   + x_2^{N_i^\infty + M_i^\Delta}y_2^{N_j^\infty } u'_2(x_2,y_2)$$
@@ -656,7 +656,7 @@ if  $\mathcal{M}(E_i)=1$ and  $E_j$ is dicritical
 $$\begin{cases}
 z^3 - x^{N^\infty_{i}}, & \mathcal{M}(E_i) \ne 1, \infty \\
 z^3 - x^{N^0_{i}}  z +  x^{N^\infty_{i}}, &  \mathcal{M}(E_i) = \infty \\
-z^3 - x_2^{\frac{N^0_i}{2}}  + x^{N_i^\infty + M_i^\Delta} , &   \mathcal{M}(E_i) = 1, \\
+z^3 - x_2^{\frac{N^0_i}{2}}z^2  + x^{N_i^\infty + M_i^\Delta} , &   \mathcal{M}(E_i) = 1, \\
 \end{cases}$$
 in the case $I= \{i\}$ 
 and 
@@ -670,6 +670,47 @@ in the case $I= \{i, j\}$.
 \end{definition}
 
 
+\begin{example} The following tables show all the trinomial forms and differential forms for the all the 0-dimensional strata and all the 1-dimensional strata with not zero Euler characteristic of Example \ref{extriplepole}
+
+
+$$\begin{array}{c|c|c|c}
+i&f_i&\omega_i& \chi()\\
+\hline
+1& z^3 + x^4z^2 + x^{13}& (2, 1, 1)&  -13\\
+2,10& z^3 + x^{10}z + x^{13}& (3, 1, 1)&1\\
+3,11& z^3 + x^{20}z + x^{26}& (5, 1, 1)& -1\\
+7,15& z^3 + \color{red}{x^{52}z} + x^{78}& (13, 1, 1)& -2\\
+8, 16&  z^3 + x^{21}z + x^{26}& (6, 1, 1)& 1\\
+ 9,17& z^3 + x^{42}z + x^{52}& (11, 1, 1)& -1\\
+\end{array}$$
+
+
+
+$$\begin{array}{c|c|c}
+(i,j)&f_{i,j}&\omega_{i,j}\\
+\hline
+&&\\
+(1, 7), (1, 15)& z^3 + x^4y^{26}z^2 + x^{13}y^{78}& (2, 13, 1)\\
+(2,3),  (10, 11)& z^3 + x^{10}y^{20}z + x^{13}y^{26}& (3, 5, 1)\\
+(3, 9), (11, 17)& z^3 + x^{20}y^{42}z + x^{26}y^{52}& (5, 11, 1)\\
+(3, 4), (11,12)& z^3 + x^{20}y^{28}z + x^{26}y^{39}& (5, 7, 1)\\
+(8, 9), (16,17)& z^3 + x^{21}y^{42}z + x^{26}y^{52}& (6, 11, 1)\\
+(4, 5), (12,13)& z^3 + x^{28}y^{36}z + x^{39}y^{52}& (7, 9, 1)\\
+(5, 6), (13,14)& z^3 + x^{36}y^{44}z + x^{52}y^{65}& (9, 11, 1)\\
+(6, 7), (14,15)& z^3 + x^{44}y^{52}z + x^{65}y^{78}& (11, 13, 1)\\
+(1, i), i=16, \dots, 28&  z^3 + x^4z^2 + x^{13}y& (2, 1, 1)\\
+(9, 31), (17, 32)& z^3 + x^{42}yz + x^{52}& (11, 1, 1)\\
+(7, 33), (15, 34)& z^3 + x^{52}z + x^{78}y& (13, 1, 1)\\
+(7, 29) (15, 30)& z^3 + x^{26}z^2 + x^{78}y& (13, 1, 1)\\
+%(12, 'f')& z^3 + x^{42}yz + x^{52}& (11, 1, 1)\\
+%(16, 'g')& z^3 + x^{52}z + x^{78}y& (13, 1, 1)\\
+%(16, 'd')& z^3 + x^{26}z^2 + x^{78}y&  (13, 1, 1)
+\end{array}$$
+
+
+\end{example}
+
+
 
 
 \section{The quasi-trinomial case}\label{3nomial}
@@ -1003,15 +1044,18 @@ The monodromy zeta function $\zeta_F(t)$ and the local topological  zeta functio
 $$(1 - t^3)^{\alpha} \prod_{I} \zeta_{f_I}(t)^{-\chi(E_I^\circ)}$$ 
 $$\sum_I \chi(E^\circ_I) Z(f_I, \omega_I)_{o_I}$$
  where $I$ varies over the statification associated to the resolution $\pi$ of $b_2b_3D_zF$, the exponent $\alpha$ is the number of points of blow-ups needed to obtain $\pi$, and $o_I$ is a point in $E^\circ_I$.
+  \end{proposition}
  
- 
- \end{proposition}
+ \begin{proof} It follows from the the formula for $\zeta_f$ in terms of a partial resolution \cite[Theorem 1]{GSLM97} and the  Stratum  principle for topological zeta functions \cite[Stratum Principle 1.2]{ABCNLM02}. Each of the $\alpha$  factors of the form   $(1-t^3)$ of $\zeta_f$ comes from the intersection of Milnor fiber of $F$ at $o$ and the line thru a center of blow-up of $\pi$. Each of these intersection sets consists of 3 points that are cyclically permuted by the monodromy. 
+ \end{proof}
  
  
 \begin{example} The resolution $\pi$ of Example \ref{extriplepole} consist of $\alpha=15$ blow-ups and the associated stratification has 35 strata. According to Proposition \ref{formestrpi} we have
 $$\zeta_F(t) = \frac{(1-t^3)^{15} \zeta^{}_{f_{2}} \zeta^{2}_{f_{4}} \zeta^{2}_{f_{5}} \zeta^{2}_{f_{6}}  \zeta^{}_{f_{8}} \zeta^{}_{f_{10}} \zeta^{2}_{f_{12}} \zeta^{2}_{f_{13}} \zeta^{2}_{f_{14}}  \zeta^{}_{f_{16}}  }
-{\zeta^{13}_{f_{1}} \zeta^{}_{f_{3}} \zeta^{4}_{f_{7}} \zeta^{}_{f_{9}} \zeta^{}_{f_{11}} \zeta^{4}_{f_{15}} \zeta^{}_{f_{17}} }$$
-$$Z(F,s)_0=-11Z(f_1,s)_0 + \sum_{i=2,8,10,16}Z(f_i,s)_0 -  \sum_{i=3, 9,11,17} Z(f_i,s)_0 - 2 \sum_{i=7,15}Z(f_i,s)_0 + \sum_{(i,j) \in S} Z(f_{i,j},s)_0,$$
+{\zeta^{13}_{f_{1}} \zeta^{}_{f_{3}} \zeta^{4}_{f_{7}} \zeta^{}_{f_{9}} \zeta^{}_{f_{11}} \zeta^{4}_{f_{15}} \zeta^{}_{f_{17}} }= \Phi^3_1 \Phi^{15}_2 \Phi_3 \Phi^2_6 \Phi^2_{13} \Phi^{15}_{26} \Phi^2_{78}$$
+where $\Phi_i$ is $i$-th the cyclotomic polymonial, and $Z(F,s)_0$ equals 
+$$-13Z(f_1,s)_0 + \sum_{i=2,8,10,16}Z(f_i,s)_0 -  \sum_{i=3, 9,11,17} Z(f_i,s)_0 - 2 \sum_{i=7,15}Z(f_i,s)_0 + \sum_{(i,j) \in S} Z(f_{i,j},s)_0=$$
+$$\frac{-1/26s^3 + 2749/6084s^2 + 1969/4056s + 1/8}{(s+1)(s+1/2)^3}$$
 where $S$ is the subset indexing the 0-dimensional strata of Example \ref{extriplepole} and we have omitted the symbols of differential forms associated to each strata for shortness sake. 
 \end{example} 
  
@@ -1223,25 +1267,37 @@ $$\frac{1}{\nu_1\nu_2(3s+1)} - \frac{2}{(3m_2s + 2\nu_1+m_2)(3n_2s + 2\nu_2+m_2)
  
  \section{Reductions}
   
-  \subsection{First reduction: Elimination of the 0-dimensional strata due to the strict transforms of $b_2$ and of $b_3$} The following Proposition shows that the o-dimensional strata associated to the strict transforms of $b_2$ and $b_3$ can eliminated from the computations of \ref{}. Each of these points can be added to the 1-dimensional strata satisfying that its clousere contains it, and the invariant is conputed with the trinomial form of the 1-dimensional strata. 
+  \subsection{First reduction: Elimination of the 0-dimensional strata due to the strict transforms of $b_2$ and of $b_3$} The following Proposition shows that the o-dimensional strata associated to the strict transforms of $b_2$ and $b_3$ can eliminated. Each of these points can be added to the 1-dimensional strata satisfying that its clousure contains it, and the invariant is computed with the trinomial form of the 1-dimensional strata. 
   
   \begin{proposition}\label{red1}Assume that $E_i$ is an exceptional divisor of the resolution $\pi$ with $\pi(E_i)=0$ (resp. $\pi(E_i)=\infty$) and that  $F^0_j$ is an strict transform of $b_2$ (resp. $b_3$) such that $E_i \cap F_j \ne \emptyset$. Then
  $$Z(F_i, \omega_i)_{o_i}=Z(F_{i,j}, \omega_{i,j})_{o_{i,j}} $$
   \end{proposition}
 
 \begin{example} The proposition \ref{red1} can be applied to the strata 
-$$E^\circ_{9,32}=E_9 \cap F^0_{32}, \quad E^\circ_{17,33}=E_{17} \cap F^0_{33}, \quad E^\circ_{7,34}= E_7 \cap F^\infty_{34} \quad \hbox{ and } \quad E^\circ_{15,35}=E_{15} \cap F^\infty_{35}$$
- of Example \ref{extriplepole}. For instance, ifrom the formulas given in the previous section;  we have 
-$$Z(F_9, \omega_9)_{o_9}=Z(F_{9,31}, \omega_{9,31})_{o_{9,31}}$$
-$$ Z(F_{15}, \omega_{15})_{o_{15}}=Z(F_{15,34}, \omega_{15,34})_{o_{15,34}}$$
+$$E^\circ_{9,31}=E_9 \cap F^0_{31}, \quad E^\circ_{17,32}=E_{17} \cap F^0_{32}, \quad E^\circ_{7,33}= E_7 \cap F^\infty_{33} \quad \hbox{ and } \quad E^\circ_{15,34}=E_{15} \cap F^\infty_{34}$$
+ of Example \ref{extriplepole}. 
+ 
+ For the strata due to the strict transforms $F^0_{31}$ and $F^0_{32}$ of $b_2$, note that $\mathcal{M}(E_9)= \mathcal{M}(E_{17})=0$. The corresponding  trinomial forms and differential forms are 
+ $$f_9=f_{17}=f_{9,31}=f_{17,32}=z^3 - x^{52}, \qquad  \omega_9=\omega_{17}=\omega_{9,31}=\omega_{17,32}= x^{11}yzdxdydz,$$
+From the formulas given in the previous section;  we have 
+$$Z(F_9, \omega_9)_{o_9}=Z(F_{9,31}, \omega_{9,31})_{o_{9,31}}= Z(F_{17}, \omega_{17})_{o_{15}}=Z(F_{17,32}, \omega_{17,32})_{o_{17,32}} = \frac{37/858s + 85/1716}{s^2 +241/156 s + 85/156}.$$
+
+ For the strata due to the strict transforms $F^\infty_{33}$ and $F^\infty_{34}$ of $b_3$, note that $\mathcal{M}(E_7)= \mathcal{M}(E_{15})=0$. The corresponding  trinomial forms and differential forms are 
+ $$f_7=f_{15}=z^3 - x^{78}, \quad f_{7,33}=f_{15,34}=z^3 - x^{52}z + x^{78}y,$$
+ and $$\omega_7=\omega_{15}=\omega_{7,33}=\omega_{15,34}= x^{13}yzdxdydz,$$
+From the formulas given in the previous section;  we have 
+$$Z(F_7, \omega_7)_{o_7}=Z(F_{7,33}, \omega_{7,33})_{o_{7,33}}= Z(F_{15}, \omega_{15})_{o_{15}}=Z(F_{15,34}, \omega_{15,34})_{o_{15,34}} = \frac{1}{26(s^2 +3/2 s + 1/2}$$
 \end{example}
 
-  \subsection{Second reduction: Elimination of connected components of exceptional divisors containing a $(-1)$-curve}
+  \subsection{Second reduction: Recursive elimination of exceptional  $(-1)$-curve not intersecting the strict transform of the discrimiant}
   
-  After performing the First reduction, connected components of exceptional divisors  
+  After performing the First reduction, exceptional $(-1)$-curves that do not intersect the strict transform of the discriminant may appear. The next proposition shows that we can eliminante these divisors.. 
   
     \begin{proposition}\label{red2} 
     \end{proposition}
+    
+    
+  \subsection{Third reduction: $\QQ$-resolution of the discriminant}
  
  \section{On generic projections}
  

From f87a6bcc5e8b3169e8b96e780147fa83995c2736 Mon Sep 17 00:00:00 2001
From: Enrique Artal <artal@unizar.es>
Date: Fri, 20 Sep 2024 12:32:12 +0200
Subject: [PATCH 2/4] actualizar antes de pull

---
 DeformacionGenerica.ipynb  |   2 +-
 DeformacionGenerica0.ipynb |   4 +-
 EjemplosGenerales.ipynb    |   4 +-
 Mult3.pdf                  | Bin 350692 -> 351403 bytes
 PruebasDeformaciones.ipynb | 456 ++++++++++++++++++++++++++++++++----
 UnLado.ipynb               | 166 ++++++++++++--
 deformaciones1.ipynb       |   4 +-
 deformaciones2.ipynb       |   2 +-
 monomios.ipynb             |   4 +-
 suspension.ipynb           |   2 +-
 zetas_triples_trenzas.py   | 458 +++++++++++++++++++++++++++++++++++++
 11 files changed, 1029 insertions(+), 73 deletions(-)
 create mode 100644 zetas_triples_trenzas.py

diff --git a/DeformacionGenerica.ipynb b/DeformacionGenerica.ipynb
index 9213037..cd9ffba 100644
--- a/DeformacionGenerica.ipynb
+++ b/DeformacionGenerica.ipynb
@@ -354,7 +354,7 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "SageMath 10.4.rc0",
+   "display_name": "SageMath 10.4.rc1",
    "language": "sage",
    "name": "sagemath-devel"
   },
diff --git a/DeformacionGenerica0.ipynb b/DeformacionGenerica0.ipynb
index 5123f8b..7c82f40 100644
--- a/DeformacionGenerica0.ipynb
+++ b/DeformacionGenerica0.ipynb
@@ -1838,7 +1838,7 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "SageMath 10.4.rc3",
+   "display_name": "SageMath 10.5.beta3",
    "language": "sage",
    "name": "sagemath-devel"
   },
@@ -1852,7 +1852,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.12.4"
+   "version": "3.12.5"
   }
  },
  "nbformat": 4,
diff --git a/EjemplosGenerales.ipynb b/EjemplosGenerales.ipynb
index aa38823..eef641a 100644
--- a/EjemplosGenerales.ipynb
+++ b/EjemplosGenerales.ipynb
@@ -673,7 +673,7 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "SageMath 10.3.rc4",
+   "display_name": "SageMath 10.4.beta6",
    "language": "sage",
    "name": "sagemath-devel"
   },
@@ -687,7 +687,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.11.1"
+   "version": "3.12.3"
   }
  },
  "nbformat": 4,
diff --git a/Mult3.pdf b/Mult3.pdf
index f02383a3b8e4a338dbf88c8c43b72a1f03eba232..b2c07243e1e916a06ae2401dfe81eb928c8dbd0b 100644
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zGEO<t=E?mc)UYl6K_Mp5MaHm4th3rM^C_g^V%TuD7Qj2Xs9*$Y;&vDB9D_bqpz9<G
z%AuU=X9_Y~QHdd(RMXx66S?->RN<4BC#avzRO3^u6?^{+Q<YEX<)9dz+T3NGpnb|_
z1)7iHnY9aJ6fXaa<t5W|<YfyV?G*qi`pHc3L~3ytP&Q8*<s{M5Jd!{6mUqrvuM0>`
zox3O+ANm=MJ0@XwZwa+NK-|+23X+Yl^C>=nCcj=Npv`Xzc?;5w!9g}SE0e&3NSU43
zB`<(qoA%$TW1=3bA4A_8C+vCHU0drBSg|%g+%HqR5coK@&@!98!-Z{WX=u8Ir^qbY
z>+|6tO854HgZ7oP7!mtg)Z$XN$QG9vp^#S$$f!fNHPqL@V7yQKsL()q0(O3|s3AjG
zMEh>>4u|@T5}Yhgau5G2zo|%Cjm&8;4lI%W5|DBVpD(m;$SeFi+kSW1&c-zi7K7P5
zr-^qbui7hE|5-GDn4H#CSW7kqPnTzVHH-5O1x+JA^gwlfvv1~|lfzTfA<0~gz|G%O
zqq-due%ApG#G`M5v&xS~&Ln$aFeW0&8J~&ZG`+j3-N7=Yx_D{_ucVJgYAP^jhGK`5
zT*?^Ipan)Bl1=#Fj`RLtM8{_^Nz7ZYKACDbE|?UTi9`vmrk)GXk=1GYv^RsV(@Kom
zvyGoQLf;#TY=rH<suViX?KXZncQ+PZ`E0^$q;fnEu~5C&;fP+TYFnM3a3Ay;J5Tk0
zB0M1>f}F6r`^0&!F#FO7!k3X>!5LJ@Zju?iwj647-f^;@QF&wa@q@9^+Q7?)gL^@z
zgWFYxfu|eiNQ{C&Dlr@9Ao=Xf>xFkHrMm&8jjPvXj;|?n8#~n+Wpy$<*d*G?X#Qzt
zpwy5I8W(0op1i#RTZ${Y9=DpN1VPXCLJLevmS49Q5Z~I0Lc{L~kw1eTLHkG!VnO^+
zi90<pT)TYYe`y{vp^Gv=b*6?N_zoWSzDOSjTlfEv=W`c)U!<)MGQh{-1rffGu(*h*
NAU-RrvaSmL{{U**FL(d|

diff --git a/PruebasDeformaciones.ipynb b/PruebasDeformaciones.ipynb
index 36e108c..8660252 100644
--- a/PruebasDeformaciones.ipynb
+++ b/PruebasDeformaciones.ipynb
@@ -18,8 +18,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 2,
-   "id": "cad50def-ecb8-403f-b50c-ce026fbd216d",
+   "execution_count": null,
+   "id": "fdc11d23-4114-49ef-828a-b7a165169c46",
    "metadata": {},
    "outputs": [],
    "source": [
@@ -28,8 +28,8 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 5,
-   "id": "5ec9dd77-8118-4439-b8cb-4c475a2a645e",
+   "execution_count": 14,
+   "id": "b67cebe5-4b5a-46be-9055-9ecbdecf96f1",
    "metadata": {},
    "outputs": [
     {
@@ -38,7 +38,7 @@
        "((0, 0, 0, 0, 0, 0, 1), (0, 0, 4, 0, 0, 0, 0), (0, 0, 0, 0, 0, 0, 3))"
       ]
      },
-     "execution_count": 5,
+     "execution_count": 14,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -53,27 +53,53 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 6,
+   "execution_count": 13,
+   "id": "dbf2120c-5214-48e1-a93f-a5079753d5f5",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "((0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 1, 1, 0, 0), (0, 0, 0, 0, 1, 0, 2))"
+      ]
+     },
+     "execution_count": 13,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Vfa, Vga, Vda = [vector(QQ, 7) for i in (1, 2, 3)]\n",
+    "Vfa[4] = Vga[3] = Vga[4] = Vda[4] = 1\n",
+    "Vda[6] = 2\n",
+    "Vfa, Vga, Vda"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 15,
    "id": "c5104650-44eb-4c9c-b016-c34994f72f38",
    "metadata": {},
    "outputs": [],
    "source": [
-    "n, Nf, Ng, Nd, Nhs, Nks, euler, nus, pols = trinomiales_general(A, Vf, Vg, (), (), Vd, total=False)"
+    "n, Nf, Ng, Nd, Nhs, Nks, euler, nus, pols = trinomiales_general(A, Vf, Vg, (), (), Vd, total=False)\n",
+    "na, Nfa, Nga, Nda, Nhsa, Nksa, eulera, nusa, polsa = trinomiales_general(A, Vfa, Vga, (), (), Vda, total=False)"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 7,
+   "execution_count": 16,
    "id": "8dccac88-2fa3-4365-990a-2571f643b9dc",
    "metadata": {},
    "outputs": [],
    "source": [
-    "zetatop, zetamon, dic = zetas(n, pols, info=True)"
+    "zetatop, zetamon, dic = zetas(n, pols, info=True)\n",
+    "zetatopa, zetamona, dica = zetas(na, polsa, info=True)"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 8,
+   "execution_count": 17,
    "id": "097214a4-61e4-4939-bad0-4e5b3c8ac716",
    "metadata": {},
    "outputs": [
@@ -81,16 +107,16 @@
      "data": {
       "text/plain": [
        "((-41/168) * (s + 16/21)^-1 * (s + 19/24)^-1 * (s + 1)^-1 * (s^2 - 283/123*s - 304/123),\n",
-       " {1: 1, 2: 1, 6: 1, 4: 1, 3: 1, 12: 1, 8: 1, 7: 1, 24: 1, 21: 1})"
+       " (-43/175) * (s + 16/21)^-1 * (s + 4/5)^-1 * (s + 1)^-1 * (s^2 - 292/129*s - 320/129))"
       ]
      },
-     "execution_count": 8,
+     "execution_count": 17,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
-    "zetatop.factor(), dic"
+    "zetatop.factor(), zetatopa.factor()"
    ]
   },
   {
@@ -129,7 +155,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 25,
+   "execution_count": 8,
    "id": "558ef7c7-39cf-47bf-b224-479f1ff365d4",
    "metadata": {},
    "outputs": [
@@ -429,27 +455,6 @@
     "    print(\"=======================\")"
    ]
   },
-  {
-   "cell_type": "code",
-   "execution_count": 26,
-   "id": "8f5e2324-c385-4a99-a4a4-96cc9aa35c12",
-   "metadata": {},
-   "outputs": [
-    {
-     "data": {
-      "text/plain": [
-       "(z^3 + x^8*z^2 + x^24*y, (11, 1, 1), 3)"
-      ]
-     },
-     "execution_count": 26,
-     "metadata": {},
-     "output_type": "execute_result"
-    }
-   ],
-   "source": [
-    "pols[6, 'd']"
-   ]
-  },
   {
    "cell_type": "code",
    "execution_count": 27,
@@ -678,6 +683,377 @@
     "dic, dic2"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": 85,
+   "id": "1eda8668-939a-49ff-9741-eecb16b95c29",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "1"
+      ]
+     },
+     "execution_count": 85,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "A2 = A1.matrix_from_rows_and_columns([0..5], [0..5])\n",
+    "A2[4, 5] = A2[5, 4] = 1\n",
+    "#A1[4, 6] = A1[6, 4] = A1[5, 6] = A1[6, 5] = 1\n",
+    "#A1[4, 4] = -3\n",
+    "A2[4, 4] = -2\n",
+    "A2[5, 5] = -1\n",
+    "A2.det()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 86,
+   "id": "f79af679-b261-4e6f-9dcb-5775bccb1c8c",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "((0, 0, 0, 0, 1, 0), (0, 0, 0, 1, 1, 0), (0, 0, 0, 0, 1, 2))"
+      ]
+     },
+     "execution_count": 86,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Vf, Vg, Vd = [vector(QQ, 6) for i in (1, 2, 3)]\n",
+    "Vf[4] = Vg[3] = Vg[4] = Vd[4] = 1\n",
+    "Vd[5] = 2\n",
+    "Vf, Vg, Vd"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 87,
+   "id": "f4f2d497-55d6-42cc-a3c1-6ee815579be8",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "n2b, Nf2b, Ng2b, Nd2b, Nhs2b, Nks2b, euler2b, nus2b, pols2b = trinomiales_general(A2, Vf, Vg, (), (), Vd, total=False)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 88,
+   "id": "6d1f647d-c103-4ffc-97da-bdcac4e0625e",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "(2, 4, 6, 7, 14, 14)\n",
+      "(3, 6, 9, 11, 21, 21)\n",
+      "(6, 12, 18, 21, 42, 44)\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(Nf2b)\n",
+    "print(Ng2b)\n",
+    "print(Nd2b)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 89,
+   "id": "27a012cc-2e37-4545-98a4-7f1ffd7f8566",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "zetatop2b, zetamon2b, dic2b = zetas(n2b, pols2b, info=True)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 90,
+   "id": "696efeb0-7276-449a-903d-c2e02cdb16f9",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "((-41/168) * (s + 16/21)^-1 * (s + 19/24)^-1 * (s + 1)^-1 * (s^2 - 283/123*s - 304/123),\n",
+       " (-39/161) * (s + 16/21)^-1 * (s + 18/23)^-1 * (s + 1)^-1 * (s^2 - 274/117*s - 32/13))"
+      ]
+     },
+     "execution_count": 90,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "zetatop.factor(), zetatop2b.factor()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 92,
+   "id": "7d299478-15d6-439a-a0ee-38557a6e303b",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "({1: 1, 2: 1, 6: 1, 4: 1, 3: 1, 12: 1, 8: 1, 7: 1, 24: 1, 21: 1},\n",
+       " {1: 1, 7: 1, 21: 1, 23: 1})"
+      ]
+     },
+     "execution_count": 92,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "dic, dic2b"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 66,
+   "id": "d236b526-d636-455a-8030-301f39e27f96",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "A = Matrix(8, [-2, 1, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0, 1, -3, 0, 1, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 1, 1, -2, 1, 0, 0, 0, 0, 0, 0, 1, -3, 0, 1, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 1, 1, -1])\n",
+    "A1 = A.matrix_from_rows_and_columns([0..6], [0..6])\n",
+    "A1[4, 5] = A1[5, 4] = 0\n",
+    "A1[4, 6] = A1[6, 4] = A1[5, 6] = A1[6, 5] = 1\n",
+    "A1[4, 4] = -3\n",
+    "A1[5, 5] = -2\n",
+    "A1[6, 6] = -1"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 73,
+   "id": "0f6ae8eb-46d6-4d8b-85ff-8b93432f88f5",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "((0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 1, 1, 0, 0), (0, 0, 0, 0, 1, 0, 1))"
+      ]
+     },
+     "execution_count": 73,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Vf, Vg, Vd = [vector(QQ, 7) for i in (1, 2, 3)]\n",
+    "Vf[4] = Vg[3] = Vg[4] = Vd[4] = Vd[6] = 1\n",
+    "Vf, Vg, Vd"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 74,
+   "id": "5d18a75f-7a9b-4ada-b28f-f79e9fb69292",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "n2a, Nf2a, Ng2a, Nd2a, Nhs2a, Nks2a, euler2a, nus2a, pols2a = trinomiales_general(A1, Vf, Vg, (), (), Vd, total=False)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 79,
+   "id": "d3af6603-a5ca-45da-8122-5e0973563b88",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "(2, 4, 6, 7, 14, 14, 28)\n",
+      "(3, 6, 9, 11, 21, 21, 42)\n",
+      "(6, 12, 18, 21, 42, 43, 86)\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(Nf2a)\n",
+    "print(Ng2a)\n",
+    "print(Nd2a)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 77,
+   "id": "4412f281-4696-4dc1-91ed-0069b58bffa0",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "zetatop2a, zetamon2a, dic2a = zetas(n2a, pols2a, info=True)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 78,
+   "id": "c39bf1b5-1ed0-4d84-8e81-5f4e4bf3fd9b",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "((-41/168) * (s + 16/21)^-1 * (s + 19/24)^-1 * (s + 1)^-1 * (s^2 - 283/123*s - 304/123),\n",
+       " (-37/154) * (s + 16/21)^-1 * (s + 17/22)^-1 * (s + 1)^-1 * (s^2 - 265/111*s - 272/111))"
+      ]
+     },
+     "execution_count": 78,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "zetatop.factor(), zetatop2a.factor()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 80,
+   "id": "441864d8-5be6-4570-8a30-b9612c0ebcb1",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "({1: 1, 2: 1, 6: 1, 4: 1, 3: 1, 12: 1, 8: 1, 7: 1, 24: 1, 21: 1},\n",
+       " {1: 1, 2: 1, 7: 1, 22: 1, 11: 1, 21: 1})"
+      ]
+     },
+     "execution_count": 80,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "dic, dic2a"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 37,
+   "id": "faed71a4-b703-4416-8c86-ce2d989999b0",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "((0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 1, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 1))"
+      ]
+     },
+     "execution_count": 37,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Vf, Vg, Vd = [vector(QQ, 8) for i in (1, 2, 3)]\n",
+    "Vf[4] = Vg[3] = Vg[4] = Vd[4] = Vd[7] = 1\n",
+    "Vf, Vg, Vd"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 38,
+   "id": "aa0267bd-3f48-4167-94f4-b6079b77cc46",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "n2, Nf2, Ng2, Nd2, Nhs2, Nks2, euler2, nus2, pols2 = trinomiales_general(A, Vf, Vg, (), (), Vd, total=False)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 39,
+   "id": "f819b1f2-2342-4951-99ee-c62a978d2308",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "(6, 12, 18, 21, 42, 42, 42, 84)\n",
+      "(6, 12, 18, 22, 42, 42, 42, 84)\n",
+      "(6, 12, 18, 21, 42, 44, 45, 90)\n"
+     ]
+    }
+   ],
+   "source": [
+    "print(3 * Nf2)\n",
+    "print(2 * Ng2)\n",
+    "print(Nd2)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 40,
+   "id": "1a340b4a-c984-46e1-a708-0c962a0912bc",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "zetatop2, zetamon2, dic2 = zetas(n2, pols2, info=True)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 41,
+   "id": "e737e117-0285-429e-8800-eab63eff90fc",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "((-41/168) * (s + 16/21)^-1 * (s + 19/24)^-1 * (s + 1)^-1 * (s^2 - 283/123*s - 304/123),\n",
+       " (-41/168) * (s + 16/21)^-1 * (s + 19/24)^-1 * (s + 1)^-1 * (s^2 - 283/123*s - 304/123))"
+      ]
+     },
+     "execution_count": 41,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "zetatop.factor(), zetatop2.factor()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 52,
+   "id": "535f3607-b56b-4557-b60b-28889a8795c4",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "({1: 1, 2: 1, 6: 1, 4: 1, 3: 1, 12: 1, 8: 1, 7: 1, 24: 1, 21: 1},\n",
+       " {1: 1, 2: 1, 6: 1, 4: 1, 3: 1, 12: 1, 8: 1, 7: 1, 24: 1, 21: 1})"
+      ]
+     },
+     "execution_count": 52,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "dic, dic2"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": 43,
@@ -768,24 +1144,24 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 48,
+   "execution_count": 49,
    "id": "0870be50-c79c-48c9-80b7-24bfd6914737",
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/plain": [
-       "((-41/168) * (s + 16/21)^-1 * (s + 19/24)^-1 * (s + 1)^-1 * (s^2 - 283/123*s - 304/123),\n",
+       "((-5/21) * (s + 16/21)^-1 * (s + 1)^-1 * (s - 16/5),\n",
        " (-5/21) * (s + 16/21)^-1 * (s + 1)^-1 * (s - 16/5))"
       ]
      },
-     "execution_count": 48,
+     "execution_count": 49,
      "metadata": {},
      "output_type": "execute_result"
     }
    ],
    "source": [
-    "zetatop.factor(), zetatop3.factor()"
+    "zetatop1.factor(), zetatop3.factor()"
    ]
   },
   {
@@ -861,7 +1237,7 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "SageMath 10.4.beta9",
+   "display_name": "SageMath 10.4.rc1",
    "language": "sage",
    "name": "sagemath-devel"
   },
diff --git a/UnLado.ipynb b/UnLado.ipynb
index d0cd61d..498c895 100644
--- a/UnLado.ipynb
+++ b/UnLado.ipynb
@@ -2,7 +2,7 @@
  "cells": [
   {
    "cell_type": "code",
-   "execution_count": 113,
+   "execution_count": 1,
    "id": "4778e32e-f561-4e4d-b63a-d4d60f91abde",
    "metadata": {},
    "outputs": [],
@@ -12,7 +12,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 115,
+   "execution_count": 2,
    "id": "f042e81b-abf8-4e09-bc67-a19d778033e6",
    "metadata": {},
    "outputs": [],
@@ -22,7 +22,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 116,
+   "execution_count": 7,
    "id": "0733b364-bbf6-486c-a5d1-e72c7a5d48ca",
    "metadata": {},
    "outputs": [],
@@ -34,7 +34,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 117,
+   "execution_count": 8,
    "id": "b81e1db9-883e-4a03-a38c-1ca296c9d25b",
    "metadata": {},
    "outputs": [],
@@ -45,7 +45,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 118,
+   "execution_count": 9,
    "id": "f97fbf75-4da5-45db-8f66-cdccdd052326",
    "metadata": {},
    "outputs": [],
@@ -60,7 +60,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 119,
+   "execution_count": 10,
    "id": "04072945-1ada-4d66-b44e-95318f073634",
    "metadata": {},
    "outputs": [],
@@ -70,7 +70,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 120,
+   "execution_count": 11,
    "id": "d9834ff4-e09f-4cf9-abd0-9b1eea6ca033",
    "metadata": {},
    "outputs": [
@@ -88,16 +88,32 @@
      },
      "metadata": {},
      "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/html": [
+       "<html>\\(\\displaystyle \\nu_{2}^{-1} \\cdot \\nu_{1}^{-1} \\cdot (3 \\mathit{ss} a_{2} + a_{2} + 3 \\nu_{2})^{-1} \\cdot (3 \\mathit{ss} a_{1} + a_{1} + 3 \\nu_{1})^{-1} \\cdot (3 \\mathit{ss} a_{1} a_{2} + a_{1} a_{2} + 3 a_{2} \\nu_{1} + 3 a_{1} \\nu_{2})\\)</html>"
+      ],
+      "text/latex": [
+       "$\\displaystyle \\nu_{2}^{-1} \\cdot \\nu_{1}^{-1} \\cdot (3 \\mathit{ss} a_{2} + a_{2} + 3 \\nu_{2})^{-1} \\cdot (3 \\mathit{ss} a_{1} + a_{1} + 3 \\nu_{1})^{-1} \\cdot (3 \\mathit{ss} a_{1} a_{2} + a_{1} a_{2} + 3 a_{2} \\nu_{1} + 3 a_{1} \\nu_{2})$"
+      ],
+      "text/plain": [
+       "nu2^-1 * nu1^-1 * (3*ss*a2 + a2 + 3*nu2)^-1 * (3*ss*a1 + a1 + 3*nu1)^-1 * (3*ss*a1*a2 + a1*a2 + 3*a2*nu1 + 3*a1*nu2)"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
     }
    ],
    "source": [
     "vertical = cono3(e1, e2, e3, V1, nu) - cono3(Q1, Q2, e3, V1, nu)\n",
-    "show(vertical.subs(sb).factor())"
+    "show(vertical.subs(sb).factor())\n",
+    "show(vertical.factor())"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 121,
+   "execution_count": 13,
    "id": "fe5aa4f3-db61-4a38-a1d1-dd100d819937",
    "metadata": {},
    "outputs": [
@@ -115,12 +131,28 @@
      },
      "metadata": {},
      "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/html": [
+       "<html>\\(\\displaystyle \\left(9\\right) \\cdot (3 \\mathit{ss} a_{2} + a_{2} + 3 \\nu_{2})^{-1} \\cdot (3 \\mathit{ss} a_{1} + a_{1} + 3 \\nu_{1})^{-1}\\)</html>"
+      ],
+      "text/latex": [
+       "$\\displaystyle \\left(9\\right) \\cdot (3 \\mathit{ss} a_{2} + a_{2} + 3 \\nu_{2})^{-1} \\cdot (3 \\mathit{ss} a_{1} + a_{1} + 3 \\nu_{1})^{-1}$"
+      ],
+      "text/plain": [
+       "(9) * (3*ss*a2 + a2 + 3*nu2)^-1 * (3*ss*a1 + a1 + 3*nu1)^-1"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
     }
    ],
    "source": [
     "horizontal = cono3(Q1, Q2, e3, V0, nu)\n",
     "horizontal = (horizontal.numerator() / 9) / (horizontal.denominator() / 9)\n",
-    "show(horizontal.subs(sb).factor())"
+    "show(horizontal.subs(sb).factor())\n",
+    "show(horizontal.factor())"
    ]
   },
   {
@@ -151,7 +183,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 123,
+   "execution_count": 14,
    "id": "29605b7d-37f7-4f22-80bd-1694134cd9b7",
    "metadata": {},
    "outputs": [
@@ -161,7 +193,7 @@
        "[9/(alpha1*alpha2), 3*a2/(alpha1*alpha2), (-3*a1)/(alpha1*alpha2)]"
       ]
      },
-     "execution_count": 123,
+     "execution_count": 14,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -172,7 +204,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 124,
+   "execution_count": 15,
    "id": "83121ff8-547c-424b-b3ce-860a80f0541e",
    "metadata": {},
    "outputs": [],
@@ -182,7 +214,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 125,
+   "execution_count": 16,
    "id": "b3963226-7397-4cb5-bdff-5382f74df8df",
    "metadata": {},
    "outputs": [
@@ -192,7 +224,7 @@
        "(3*a1/alpha1, 3*a1/alpha1, 3*a2/alpha2, 3*a2/alpha2, (a1 + 3*nu1)/alpha1)"
       ]
      },
-     "execution_count": 125,
+     "execution_count": 16,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -203,7 +235,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 126,
+   "execution_count": 17,
    "id": "36201bc7-a7e2-4c9e-a8e9-5865cb9686f9",
    "metadata": {},
    "outputs": [],
@@ -214,7 +246,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 127,
+   "execution_count": 18,
    "id": "d77f48f5-2236-4cb6-a211-14f3623539c0",
    "metadata": {},
    "outputs": [
@@ -241,7 +273,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 128,
+   "execution_count": 19,
    "id": "c18631c1-f18a-48a2-8e9a-727d347ef89a",
    "metadata": {},
    "outputs": [
@@ -259,12 +291,102 @@
      },
      "metadata": {},
      "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/html": [
+       "<html>\\(\\displaystyle \\left(-1\\right) \\cdot \\nu_{2}^{-1} \\cdot \\nu_{1}^{-1} \\cdot (\\mathit{ss} + 1)^{-1} \\cdot (3 \\mathit{ss} a_{2} + a_{2} + 3 \\nu_{2})^{-1} \\cdot (3 \\mathit{ss} a_{1} + a_{1} + 3 \\nu_{1})^{-1} \\cdot (3 \\mathit{ss} \\nu_{1} \\nu_{2} \\mathit{cd}^{2} - 3 \\mathit{ss}^{2} a_{1} a_{2} - 4 \\mathit{ss} a_{1} a_{2} - 3 \\mathit{ss} a_{2} \\nu_{1} - 3 \\mathit{ss} a_{1} \\nu_{2} - 9 \\mathit{ss} \\nu_{1} \\nu_{2} - a_{1} a_{2} - 3 a_{2} \\nu_{1} - 3 a_{1} \\nu_{2} - 9 \\nu_{1} \\nu_{2})\\)</html>"
+      ],
+      "text/latex": [
+       "$\\displaystyle \\left(-1\\right) \\cdot \\nu_{2}^{-1} \\cdot \\nu_{1}^{-1} \\cdot (\\mathit{ss} + 1)^{-1} \\cdot (3 \\mathit{ss} a_{2} + a_{2} + 3 \\nu_{2})^{-1} \\cdot (3 \\mathit{ss} a_{1} + a_{1} + 3 \\nu_{1})^{-1} \\cdot (3 \\mathit{ss} \\nu_{1} \\nu_{2} \\mathit{cd}^{2} - 3 \\mathit{ss}^{2} a_{1} a_{2} - 4 \\mathit{ss} a_{1} a_{2} - 3 \\mathit{ss} a_{2} \\nu_{1} - 3 \\mathit{ss} a_{1} \\nu_{2} - 9 \\mathit{ss} \\nu_{1} \\nu_{2} - a_{1} a_{2} - 3 a_{2} \\nu_{1} - 3 a_{1} \\nu_{2} - 9 \\nu_{1} \\nu_{2})$"
+      ],
+      "text/plain": [
+       "(-1) * nu2^-1 * nu1^-1 * (ss + 1)^-1 * (3*ss*a2 + a2 + 3*nu2)^-1 * (3*ss*a1 + a1 + 3*nu1)^-1 * (3*ss*nu1*nu2*cd^2 - 3*ss^2*a1*a2 - 4*ss*a1*a2 - 3*ss*a2*nu1 - 3*ss*a1*nu2 - 9*ss*nu1*nu2 - a1*a2 - 3*a2*nu1 - 3*a1*nu2 - 9*nu1*nu2)"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
     }
    ],
    "source": [
     "prueba = horizontal - arista + vertical\n",
     "# prueba.numerator().variables() + prueba.denominator().variables()\n",
-    "show(prueba.subs(sb).factor())"
+    "show(prueba.subs(sb).factor())\n",
+    "show(prueba.factor())"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 23,
+   "id": "5c36c8a4-f5f0-43d8-bbfc-35e889012f6e",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<html>\\(\\displaystyle 3 \\mathit{ss} \\nu_{1} \\nu_{2} \\mathit{cd}^{2} - 3 \\mathit{ss}^{2} a_{1} a_{2} - 4 \\mathit{ss} a_{1} a_{2} - 3 \\mathit{ss} a_{2} \\nu_{1} - 3 \\mathit{ss} a_{1} \\nu_{2} - 9 \\mathit{ss} \\nu_{1} \\nu_{2} - a_{1} a_{2} - 3 a_{2} \\nu_{1} - 3 a_{1} \\nu_{2} - 9 \\nu_{1} \\nu_{2}\\)</html>"
+      ],
+      "text/latex": [
+       "$\\displaystyle 3 \\mathit{ss} \\nu_{1} \\nu_{2} \\mathit{cd}^{2} - 3 \\mathit{ss}^{2} a_{1} a_{2} - 4 \\mathit{ss} a_{1} a_{2} - 3 \\mathit{ss} a_{2} \\nu_{1} - 3 \\mathit{ss} a_{1} \\nu_{2} - 9 \\mathit{ss} \\nu_{1} \\nu_{2} - a_{1} a_{2} - 3 a_{2} \\nu_{1} - 3 a_{1} \\nu_{2} - 9 \\nu_{1} \\nu_{2}$"
+      ],
+      "text/plain": [
+       "3*ss*nu1*nu2*cd^2 - 3*ss^2*a1*a2 - 4*ss*a1*a2 - 3*ss*a2*nu1 - 3*ss*a1*nu2 - 9*ss*nu1*nu2 - a1*a2 - 3*a2*nu1 - 3*a1*nu2 - 9*nu1*nu2"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "uuu= (ss + 1) * ((3 * ss + 1) * a1 * a2 + 3 * (nu1 * a2 + nu2 * a1 + 3 * nu1 * nu2)) - 3 * nu1 * nu2 * cd^2 *ss\n",
+    "show(-uuu)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 28,
+   "id": "52254bac-4ce0-473f-a6c8-9853e4180df9",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "-3 \\mathit{ss} \\nu_{1} \\nu_{2} \\mathit{cd}^{2} + 3 \\mathit{ss}^{2} a_{1} a_{2} + 4 \\mathit{ss} a_{1} a_{2} + 3 \\mathit{ss} a_{2} \\nu_{1} + 3 \\mathit{ss} a_{1} \\nu_{2} + 9 \\mathit{ss} \\nu_{1} \\nu_{2} + a_{1} a_{2} + 3 a_{2} \\nu_{1} + 3 a_{1} \\nu_{2} + 9 \\nu_{1} \\nu_{2}"
+      ]
+     },
+     "execution_count": 28,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "latex(uuu)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 27,
+   "id": "b4f731bf-7872-4e7f-a8dc-f1a7666fcd31",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<html>\\(\\displaystyle \\left(-1\\right) \\cdot \\nu_{1}^{-1} \\cdot (\\mathit{ss} + 1)^{-1} \\cdot (3 \\mathit{ss} a_{1} + a_{1} + 3 \\nu_{1})^{-1} \\cdot (\\mathit{ss} \\nu_{1} \\mathit{cd}^{2} - \\mathit{ss} a_{1} - 3 \\mathit{ss} \\nu_{1} - a_{1} - 3 \\nu_{1})\\)</html>"
+      ],
+      "text/latex": [
+       "$\\displaystyle \\left(-1\\right) \\cdot \\nu_{1}^{-1} \\cdot (\\mathit{ss} + 1)^{-1} \\cdot (3 \\mathit{ss} a_{1} + a_{1} + 3 \\nu_{1})^{-1} \\cdot (\\mathit{ss} \\nu_{1} \\mathit{cd}^{2} - \\mathit{ss} a_{1} - 3 \\mathit{ss} \\nu_{1} - a_{1} - 3 \\nu_{1})$"
+      ],
+      "text/plain": [
+       "(-1) * nu1^-1 * (ss + 1)^-1 * (3*ss*a1 + a1 + 3*nu1)^-1 * (ss*nu1*cd^2 - ss*a1 - 3*ss*nu1 - a1 - 3*nu1)"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "show(prueba(a2=0, nu2=1).factor())"
    ]
   },
   {
@@ -944,9 +1066,9 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "SageMath 10.3.rc3",
+   "display_name": "SageMath 10.4.beta6",
    "language": "sage",
-   "name": "sagemath-310"
+   "name": "sagemath-devel"
   },
   "language_info": {
    "codemirror_mode": {
@@ -958,7 +1080,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.11.1"
+   "version": "3.12.3"
   }
  },
  "nbformat": 4,
diff --git a/deformaciones1.ipynb b/deformaciones1.ipynb
index 0b24641..e623dc5 100644
--- a/deformaciones1.ipynb
+++ b/deformaciones1.ipynb
@@ -723,7 +723,7 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "SageMath 10.4.beta3",
+   "display_name": "SageMath 10.4.rc1",
    "language": "sage",
    "name": "sagemath-devel"
   },
@@ -737,7 +737,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.12.2"
+   "version": "3.12.3"
   }
  },
  "nbformat": 4,
diff --git a/deformaciones2.ipynb b/deformaciones2.ipynb
index 2d44437..51d12ab 100644
--- a/deformaciones2.ipynb
+++ b/deformaciones2.ipynb
@@ -488,7 +488,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.12.2"
+   "version": "3.12.3"
   }
  },
  "nbformat": 4,
diff --git a/monomios.ipynb b/monomios.ipynb
index ebb4331..db6c3e2 100644
--- a/monomios.ipynb
+++ b/monomios.ipynb
@@ -468,7 +468,7 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "SageMath 10.4.beta3",
+   "display_name": "SageMath 10.4.beta6",
    "language": "sage",
    "name": "sagemath-devel"
   },
@@ -482,7 +482,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.12.2"
+   "version": "3.12.3"
   }
  },
  "nbformat": 4,
diff --git a/suspension.ipynb b/suspension.ipynb
index b136796..11d2345 100644
--- a/suspension.ipynb
+++ b/suspension.ipynb
@@ -987,7 +987,7 @@
  ],
  "metadata": {
   "kernelspec": {
-   "display_name": "SageMath 10.4.beta3",
+   "display_name": "SageMath 10.4.beta4",
    "language": "sage",
    "name": "sagemath-devel"
   },
diff --git a/zetas_triples_trenzas.py b/zetas_triples_trenzas.py
new file mode 100644
index 0000000..3724e38
--- /dev/null
+++ b/zetas_triples_trenzas.py
@@ -0,0 +1,458 @@
+def trinomiales_general(A, Vf, Vg, Vhs, Vd, total=True):
+    r"""
+    Return data necessary to compute the zeta functions of
+    a triple singularity defined by
+
+        .. math::
+            z^3 -3 \prod_{i=1}^h h_i^{n_i} f z^2
+            + \prod_{i=1}^h h_i g,
+
+    where :math:`f, g, h_i` are polynomials in two variables,
+    and :math:`n_i \geq 1`.
+
+    We consider an embedded resolution of :math:`fg(f^3 - g^2)` with
+    a stratification of the exceptional divisor:
+
+    - :math:`\check{E}_i`: the smooth part of the total divisor
+
+    - :math:`E_i\cap E_j`: the double points of the divisor
+
+    - :math:`E_i\cap \{f=0\}`: the intersections with the strict
+      transform of :math:`f`
+
+    - :math:`E_i\cap \{g=0\}`: the intersections with the strict
+      transform of :math:`g`
+
+    - :math:`E_i\cap \{f^3=g^2\}`: the intersections with the strict
+      transform of :math:`f^3-g^2`
+
+    It is also possible to forget the strict transforms of :math:`f, g`
+    and even to consider only resolutions of :math:`f^3 - g^2`.
+
+    INPUT:
+
+    - ``A`` -- the incidence matrix of a resolution
+
+    - ``Vf`` -- the vector with the intersection numbers of the strict
+      transform of :math:`f` with the divisors of the resolution
+
+    - ``Vg`` -- the vector with the intersection numbers of the strict
+      transform of :math:`g` with the divisors of the resolution
+
+    - ``Vhs`` -- a list of tuple whose elements are tuples `(h_i, n_i)`
+
+
+    - ``Vd`` -- the vector with the intersection numbers of the strict
+      transform of :math:`f^3 - g^2` with the divisors of the resolution
+
+    - ``total`` -- boolean (default: ``True``), whether we consider also
+      the strata corresponding to the strict transforms of :math:`f, g`.
+
+    OUTPUT:
+
+    Polynomials :math:`z^3 + M_1 z + M_0` or :math:`z^3 + M_2 z^2 + M_0` are
+    assigned to each strata. The monomials are of the form :math:`x^k` for the
+    smooth parts of the divisors and :math:`x^k y^l` for the double points.
+    Each variable is associated to a divisor or to a strict transform
+
+    - ``n`` -- the number of irreducible components of the exceptional divisor
+
+    - ``Nf`` -- the vector in :math:`\mathbb{Z}^n` of the multiplicities for
+      the total transform of :math:`f`
+
+    - ``Ng`` -- the vector in :math:`\mathbb{Z}^n` of the multiplicities for
+      the total transform of :math:`g`
+
+    - ``Nhs`` -- a list of tuple whose elements are tuples, the first element
+      is the vector in :math:`\mathbb{Z}^n` of the multiplicities for
+      the total transform of :math:`h_i` and the second is :math:`n_i`
+
+    - ``Nd`` -- the vector in :math:`\mathbb{Z}^n` of the multiplicities for
+      the total transform of :math:`f^3 - g^2`
+
+    - ``euler`` -- the vector in :math:`\mathbb{Z}^n` of the Euler
+      characteristic of the smooth part of the strata. If ``total=True`` this
+      smooth part does not contain the intersections with total transform
+      of :math:`f, g`
+
+    - ``nus`` -- the vector in :math:`\mathbb{Z}^n` of the canonical
+      multiplicities
+
+    - ``pols`` -- a dictionary with keys ``(i, )``, ``(i, j)``, ``(i, 'f')``
+      (if ``total=True``), ``(i, 'g')`` (if ``total=True``), and ``(i, 'd')``
+      assigning the polynomial of the stratum (non-degenerated with respect
+      to its Newton polytope), the weights ``(nui, nuj, 1)`` coding
+      the form
+      :math:`x^{\nu_i-1}y^{\nu_j-1}\operatorname{d}\! x \operatorname{d}\! y \operatorname{d}\! z`
+
+    EXAMPLES::
+
+        sage: R.<x, y, z> = QQ[]
+        sage: f, g = (x, y)
+        sage: A = Matrix(QQ, 3, [-3, 0, 1, 0, -2, 1, 1, 1, -1])
+        sage: Vf = vector(QQ, [1, 0, 0])
+        sage: Vg = vector(QQ, [0, 1, 0])
+        sage: Vd = vector(QQ, [0, 0, 1])
+        sage: trinomiales_general(A, Vf, Vg, (), Vd)
+        (3, (1, 1, 2), (1, 2, 3), (2, 3, 6), (0, 0, -1), (2, 3, 5),
+         {(0, 'f'): (x + x*y*z + z^3, (2, 1, 1), 1),
+          (0, 2): (z^3 + x*y^3 + x*y^2*z, (2, 5, 1), 1),
+          (1, 'g'): (x*z + x^2*y + z^3, (3, 1, 1), 1),
+          (1, 2): (z^3 + x*y^2*z + x^2*y^3, (3, 5, 1), 1),
+          (2, 'd'): (x*z^2 + z^3 + x^3*y, (5, 1, 1), 1),
+          0: (x + x*z + z^3, (2, 1, 1), 0),
+          1: (x^2 + x*z + z^3, (3, 1, 1), 0),
+          2: (x^3 + x^2*z + z^3, (5, 1, 1), -1)})
+        sage: trinomiales(A, Vf, Vg, Vd, (), total=False)
+        (3, (1, 1, 2), (1, 2, 3), (2, 3, 6), (1, 1, -1), (2, 3, 5),
+         {(0, 2): (z^3 + x*y^3 + x*y^2*z, (2, 5, 1), 1),
+          (1, 2): (z^3 + x*y^2*z + x^2*y^3, (3, 5, 1), 1),
+          (2, 'd'): (x*z^2 + z^3 + x^3*y, (5, 1, 1), 1),
+          0: (x + x*z + z^3, (2, 1, 1), 1),
+          1: (x^2 + x*z + z^3, (3, 1, 1), 1),
+          2: (x^3 + x^2*z + z^3, (5, 1, 1), -1)})
+    """
+    R = PolynomialRing(QQ, 'x, y, z', order='negdeglex')
+    x, y, z = R.gens()
+    n = A.nrows()
+    A1 = -A ** -1
+    Nf, Ng, Nd = [A1 * N0 for N0 in (Vf, Vg, Vd)]
+    Nhs = [(A1 * N0, h) for N0, h in Vhs]
+    ch = len(Nhs)
+    NNf = Nf + sum(j * V0 for V0, j in Nhs)
+    NNg = Ng + sum(V0 for V0, j in Nhs)
+    NNd = Nd + 2 * sum(V0 for V0, j in Nhs)
+    diag = vector(A.diagonal())
+    unos = vector(n * [1])
+    if total:
+        VV = Vf + Vg
+    else:
+        VV = Vf.parent()(0)
+    resta = VV + Vd + sum(V0 for V0, m in Vhs)
+    euler = (-A + 2 * identity_matrix(n)) * unos + diag - resta
+    nus = A1 * (diag + 2 * unos) + unos
+    expos = {}
+    for j in range(n):
+        a, b = (NNf[j], NNg[j])
+        eq = 3 * a == 2 * NNg[j]
+        nnd = min(3 * a, 2 * b) == NNd[j]
+        if nnd:
+            expos[j] = ((a, b), (a / 2, b), nus[j], eq, nnd)
+        else:
+            expos[j] = ((a, b), (a / 2, NNd[j] - b), nus[j], eq, nnd)
+    cadenas = ('d', )
+    cadenash = tuple(('h', h) for h in range(ch))
+    if total:
+        expos['f'] = ((1, 0), (0, 0), 1, False, True)
+        expos['g'] = ((0, 1), (0, 0), 1, False, True)
+        cadenas = ('f', 'g') + cadenas
+    expos['d'] = ((0, 0), (0, 1), 1, True, False)
+    for h, V0 in enumerate(Vhs):
+        expos['h', h] = ((V0[1], 1), (0, 0), 1, False, True)
+    pols = {}
+    for j in range(n):
+        if expos[j][-1]:
+            a, b = expos[j][0]
+            pols[j] = (z ** 3 + x ** a * z + x ** b, (expos[j][2], 1, 1),
+                       euler[j])
+        else:
+            a, b = expos[j][1]
+            pols[j] = (z ** 3 + x ** a * z ** 2 + x ** b,
+                       (expos[j][2], 1, 1), euler[j])
+    interdiv = [(i, j) for i, j in Combinations(n, 2) if A[i, j] != 0]
+    interstrict = [(i, j) for i in range(n) for j in cadenas
+                   if eval('V' + j)[i] != 0]
+    interh = [(i, j) for i in range(n) for j in cadenash if Vhs[j[1]][0][i] != 0]
+    for i, j in interdiv + interstrict + interh:
+        if isinstance(j, str):
+            ep = eval('V' + j)[i]
+        else:
+            ep = 1
+        eqi, nndi = expos[i][-2:]
+        eqj, nndj = expos[j][-2:]
+        nd = eqi and eqj and not (nndi and nndj)
+        if not nd:
+            ai, bi = expos[i][0]
+            aj, bj = expos[j][0]
+            pols[i, j] = (z ** 3 + x ** ai * y ** aj * z + x ** bi * y ** bj,
+                          (expos[i][2], expos[j][2], 1), ep)
+        else:
+            ai, bi = expos[i][1]
+            aj, bj = expos[j][1]
+            pols[i, j] = (z ** 3 + x ** ai * y ** aj * z ** 2 +
+                          x ** bi * y ** bj, (expos[i][2], expos[j][2], 1), ep)
+    return n, Nf, Ng, Nd, Nhs, euler, nus, pols
+
+def exp_trenza_dis(tau):
+    B = tau.parent()
+    m = tau.exponent_sum()
+    a, b = m.quo_rem(6)
+    no = False
+    while not no:
+      no = bool(tau.conjugating_braid(B.delta()**(2 * a) * B.gen(0)**b))
+      if not no:
+        a, b = a - 1, b + 6
+    return (2 * a, None, 3 * a + b)
+
+def exp_trenza(tau):
+    B = tau.parent()
+    m = tau.exponent_sum()
+    if m.gcd(6) == 1:
+      return exp_trenza_dis(tau)
+    elif m.gcd(6) == 3:
+      polo = bool(tau.conjugating_braid(B.delta()**(m / 3)))
+      if polo:
+        return (None, m / 3, None)
+      else:
+        return exp_trenza_dis(tau)
+    elif m.gcd(6) == 2:
+      cero = bool(tau.conjugating_braid(B([1, 2])**(m / 2)))
+      if cero:
+        return (None, None, m / 2)
+      else:
+        return exp_trenza_dis(tau)
+    else:
+      ct = bool(tau.conjugating_braid(B.delta()**(m / 3)))
+      if ct:
+        return (m / 6, m / 3, m / 2)
+      else:
+        return exp_trenza_dis(tau)
+
+
+
+
+def trinomiales_trenza(A, trenzas, Vhs, Vd):
+    r"""
+    Return data necessary to compute the zeta functions of
+    a triple singularity defined by
+
+        .. math::
+            z^3 -3 \prod_{i=1}^h h_i^{n_i} f z^2
+            + \prod_{i=1}^h h_i g,
+
+    where :math:`f, g, h_i` are polynomials in two variables,
+    and :math:`n_i \geq 1`.
+
+    We consider an embedded resolution of :math:`fg(f^3 - g^2)` with
+    a stratification of the exceptional divisor:
+
+    - :math:`\check{E}_i`: the smooth part of the total divisor
+
+    - :math:`E_i\cap E_j`: the double points of the divisor
+
+    - :math:`E_i\cap \{f=0\}`: the intersections with the strict
+      transform of :math:`f`
+
+    - :math:`E_i\cap \{g=0\}`: the intersections with the strict
+      transform of :math:`g`
+
+    - :math:`E_i\cap \{f^3=g^2\}`: the intersections with the strict
+      transform of :math:`f^3-g^2`
+
+    It is also possible to forget the strict transforms of :math:`f, g`
+    and even to consider only resolutions of :math:`f^3 - g^2`.
+
+    INPUT:
+
+    - ``A`` -- the incidence matrix of a resolution
+
+    - ``Vf`` -- the vector with the intersection numbers of the strict
+      transform of :math:`f` with the divisors of the resolution
+
+    - ``Vg`` -- the vector with the intersection numbers of the strict
+      transform of :math:`g` with the divisors of the resolution
+
+    - ``Vhs`` -- a list of tuple whose elements are tuples `(h_i, n_i)`
+
+
+    - ``Vd`` -- the vector with the intersection numbers of the strict
+      transform of :math:`f^3 - g^2` with the divisors of the resolution
+
+    - ``total`` -- boolean (default: ``True``), whether we consider also
+      the strata corresponding to the strict transforms of :math:`f, g`.
+
+    OUTPUT:
+
+    Polynomials :math:`z^3 + M_1 z + M_0` or :math:`z^3 + M_2 z^2 + M_0` are
+    assigned to each strata. The monomials are of the form :math:`x^k` for the
+    smooth parts of the divisors and :math:`x^k y^l` for the double points.
+    Each variable is associated to a divisor or to a strict transform
+
+    - ``n`` -- the number of irreducible components of the exceptional divisor
+
+    - ``Nf`` -- the vector in :math:`\mathbb{Z}^n` of the multiplicities for
+      the total transform of :math:`f`
+
+    - ``Ng`` -- the vector in :math:`\mathbb{Z}^n` of the multiplicities for
+      the total transform of :math:`g`
+
+    - ``Nhs`` -- a list of tuple whose elements are tuples, the first element
+      is the vector in :math:`\mathbb{Z}^n` of the multiplicities for
+      the total transform of :math:`h_i` and the second is :math:`n_i`
+
+    - ``Nd`` -- the vector in :math:`\mathbb{Z}^n` of the multiplicities for
+      the total transform of :math:`f^3 - g^2`
+
+    - ``euler`` -- the vector in :math:`\mathbb{Z}^n` of the Euler
+      characteristic of the smooth part of the strata. If ``total=True`` this
+      smooth part does not contain the intersections with total transform
+      of :math:`f, g`
+
+    - ``nus`` -- the vector in :math:`\mathbb{Z}^n` of the canonical
+      multiplicities
+
+    - ``pols`` -- a dictionary with keys ``(i, )``, ``(i, j)``, ``(i, 'f')``
+      (if ``total=True``), ``(i, 'g')`` (if ``total=True``), and ``(i, 'd')``
+      assigning the polynomial of the stratum (non-degenerated with respect
+      to its Newton polytope), the weights ``(nui, nuj, 1)`` coding
+      the form
+      :math:`x^{\nu_i-1}y^{\nu_j-1}\operatorname{d}\! x \operatorname{d}\! y \operatorname{d}\! z`
+
+    EXAMPLES::
+
+        sage: R.<x, y, z> = QQ[]
+        sage: f, g = (x, y)
+        sage: A = Matrix(QQ, 3, [-3, 0, 1, 0, -2, 1, 1, 1, -1])
+        sage: Vf = vector(QQ, [1, 0, 0])
+        sage: Vg = vector(QQ, [0, 1, 0])
+        sage: Vd = vector(QQ, [0, 0, 1])
+        sage: trinomiales_general(A, Vf, Vg, (), Vd)
+        (3, (1, 1, 2), (1, 2, 3), (2, 3, 6), (0, 0, -1), (2, 3, 5),
+         {(0, 'f'): (x + x*y*z + z^3, (2, 1, 1), 1),
+          (0, 2): (z^3 + x*y^3 + x*y^2*z, (2, 5, 1), 1),
+          (1, 'g'): (x*z + x^2*y + z^3, (3, 1, 1), 1),
+          (1, 2): (z^3 + x*y^2*z + x^2*y^3, (3, 5, 1), 1),
+          (2, 'd'): (x*z^2 + z^3 + x^3*y, (5, 1, 1), 1),
+          0: (x + x*z + z^3, (2, 1, 1), 0),
+          1: (x^2 + x*z + z^3, (3, 1, 1), 0),
+          2: (x^3 + x^2*z + z^3, (5, 1, 1), -1)})
+        sage: trinomiales(A, Vf, Vg, Vd, (), total=False)
+        (3, (1, 1, 2), (1, 2, 3), (2, 3, 6), (1, 1, -1), (2, 3, 5),
+         {(0, 2): (z^3 + x*y^3 + x*y^2*z, (2, 5, 1), 1),
+          (1, 2): (z^3 + x*y^2*z + x^2*y^3, (3, 5, 1), 1),
+          (2, 'd'): (x*z^2 + z^3 + x^3*y, (5, 1, 1), 1),
+          0: (x + x*z + z^3, (2, 1, 1), 1),
+          1: (x^2 + x*z + z^3, (3, 1, 1), 1),
+          2: (x^3 + x^2*z + z^3, (5, 1, 1), -1)})
+    """
+    R = PolynomialRing(QQ, 'x, y, z', order='negdeglex')
+    x, y, z = R.gens()
+    n = A.nrows()
+    A1 = -A ** -1
+    Nd = A1 * Vd
+    Nhs = [(A1 * N0, h) for N0, h in Vhs]
+    ch = len(Nhs)
+    NNd = Nd + 2 * sum(V0 for V0, j in Nhs)
+    diag = vector(A.diagonal())
+    unos = vector(n * [1])
+    resta = Vd + sum(V0 for V0, m in Vhs)
+    euler = (-A + 2 * identity_matrix(n)) * unos + diag - resta
+    nus = A1 * (diag + 2 * unos) + unos
+    expos = {j: exp_trenza(trenzas[j]) for j in range(n)}
+    cadenas = ('d', )
+    cadenash = tuple(('h', h) for h in range(ch))
+    expos['d'] = (0, 0, 1)
+    for h, V0 in enumerate(Vhs):
+        expos['h', h] = (0, V0[1], 1)
+    pols = {}
+    for j in range(n):
+        p = z**3
+        for k, e in enumerate(expos[j]):
+            if e:
+                p += x**e * z**(2 - k)
+        pols[j] = (p, (nus[j], 1, 1), euler[j])
+    interdiv = [(i, j) for i, j in Combinations(n, 2) if A[i, j] != 0]
+    interstrict = [(i, j) for i in range(n) for j in cadenas
+                   if eval('V' + j)[i] != 0]
+    interh = [(i, j) for i in range(n) for j in cadenash if Vhs[j[1]][0][i] != 0]
+    for i, j in interdiv + interstrict + interh:
+        if isinstance(j, str):
+            ep = eval('V' + j)[i]
+            nu = (nus[i], 1, 1)
+        else:
+            ep = 1
+            nu = (nus[i], nus[j], 1)
+        p = z**3
+        for k, (e1, e2) in enumerate(zip(expos[i], expos[j])):
+            if e1 and e2:
+                p += x**e1 * y**e2 * z**(2 - k)
+        pols[i, j] = (p, nu, ep)
+    return n, Nd, Nhs, euler, nus, pols
+
+def estrato(pol, pesos, euler, info=False):
+    r"""
+    Return the topological and Igusa zeta functions associated to a stratum
+
+    INPUT:
+
+    - ``pol`` -- a trinomial non degenerated with respect to its Newton
+      polyhedron
+
+    - ``pesos`` -- the canonical weights :math:`\nu:i`
+
+    - `euler`` -- the Euler characteristic of the stratum
+
+    - ``info`` -- boolean (default: False), if ``True`` a dictionary giving
+      the powers of cyclotomic polynomials for the monodromy zeta function
+      is given
+
+    EXAMPLES::
+
+        sage: R.<x, y, z> = QQ[]
+        sage: F = z**3 + x**4 * y**6 * z + x**7 * y**9
+        sage: estrato(F, (1, 1, 1), 1, info=True)
+        ((1/3*s**2 + 1/2*s + 2/9)/(s^3 + 35/18*s^2 + 7/6*s + 2/9), -t^3 + 1,
+         {1: 1, 3: 1})
+    """
+    Z1 = ZetaFunctions(pol)
+    zetatop = euler * Z1.topological_zeta(local=True, weights=pesos)
+    zetamon = Z1.monodromy_zeta() ** euler
+    if info:
+        dic = {}
+        for p, j in zetamon.factor():
+            c = p.is_cyclotomic(certificate=True)
+            dic[c] = j
+        return zetatop, zetamon, dic
+    return zetatop, zetamon
+
+
+def zetas(n, pols, info=False):
+    r"""
+    Return the topological and Igusa zeta functions from the data of
+    a stratification
+
+    INPUT:
+
+    - ``n`` -- the number of exceptional divisors
+
+    - ``pols`` -- a dictionary from ``trinomiales``
+
+    - ``info`` -- boolean (default: False), if ``True`` a dictionary giving
+      the powers of cyclotomic polynomials for the monodromy zeta function
+      is given
+
+    EXAMPLES::
+
+        sage: A = Matrix(QQ, 3, [-3, 0, 1, 0, -2, 1, 1, 1, -1])
+        sage: Vf = vector(QQ, [1, 0, 0])
+        sage: Vg = vector(QQ, [0, 1, 0])
+        sage: Vd = vector(QQ, [0, 0, 1])
+        sage: n, Nf, Ng, Nd, euler, nus, pols = trinomiales(A, Vf, Vg, Vd)
+        sage: zetas(n, pols, info=True)
+        (1/(s + 1), -t + 1, {1: 1})
+    """
+    zetatop = 0
+    T = PolynomialRing(QQ, 't')
+    t = T.gen(0)
+    zetamon = 1 / (1 - t ** 3) ** n
+    for k in pols:
+        zt, zm = estrato(*pols[k])
+        zetatop += zt
+        zetamon *= zm
+    if info:
+        dic = {}
+        for p, j in zetamon.factor():
+            c = p.is_cyclotomic(certificate=True)
+            dic[c] = j
+        return zetatop, zetamon, dic
+    return zetatop, zetamon

From 6f79b3ace61b75382d893dd15bc9d202405f68f9 Mon Sep 17 00:00:00 2001
From: "m.gonzalez" <m.gonzalez@unizar.es>
Date: Fri, 20 Sep 2024 16:29:28 +0200
Subject: [PATCH 3/4] actualizar Mult3.tex

---
 Mult3.tex | 30 ++++++++++++++++++------------
 1 file changed, 18 insertions(+), 12 deletions(-)

diff --git a/Mult3.tex b/Mult3.tex
index 43b4554..08a324c 100644
--- a/Mult3.tex
+++ b/Mult3.tex
@@ -391,14 +391,17 @@ $$ \prod_{i=1}^n h_i^{2} \prod_{j=1}^m k_j^3 \left (  \prod_{i=1}^n h_i^{3n_i-2}
 
 
 \begin{proposition}
-The singular point is isolated if and only if the disciminant $D_zF$ is reduced.
+The singular point is isolated if and only if the discriminant $D_zF$ is reduced.
 \end{proposition}
 
+\noindent \textcolor{red}{Proof. Ask Enrique.}
 
-\begin{proposition}\label{prop.non-deg}
-The monic Weierstrass polynomial $z^3 -3b_2z + 2b_3$ is non degenerate unless $M_1(c_1+A)$ and $b_3 = M_2(c_2 + B)$ with $M_1^3 = M_2^2$ and \textcolor{red}{$2c_1=3c_2$}
+
+\begin{proposition}\label{prop.non-deg} Let $b_2, b_3 \in \CC[[x, y]]$ with $b_2(0), b_3(0) = 0$. Let us write  $b_2= M_2(c_2+A)$ and $b_3 = M_3(c_3 + B)$ with $M_2, M_3 \in \CC[x,y]$ (monic) monomials, $c_2, c_3 \in \CC^\ast$ and $A, B \in \CC[[x,y]]$ such that $A(0), B(0) = 0$. 
+The monic Weierstrass polynomial $z^3 -3b_2z + 2b_3$ is non degenerate unless  $(c_2M_2)^3 = (c_3M_3)^2$.
 \end{proposition}
 
+\noindent \textcolor{blue}{Proof. The Newton polyhedron of  $z^3 -3b_2z + 2b_3$ can be 2-dimensional, polygonal with two segments or polygonal with a unique segment. In the two first cases, it is non degenerate. The compact faces of the Newton polyhedron of $z^3 -3b_2z + 2b_3$ lie on a segment  if and only if  $(M_2)^3 = (M_3)^2$. The condition of $z^3 -3b_2z + 2b_3$ being non degenerate  is equivalent to $z^3 -3b_2z + 2b_3 \in \CC[z]$ having a double root. Notice a triple root is not possible because the degree 2 monomial is zero. Assume that $z^3 -3b_2z + 2b_3 =(z-a)^2(z-b)$ for some $a, b \in \CC^\ast$. Equivalently, we have $2a+b=0$, and therefor  $c_2=(\frac{b}{2})^2}$ and $c_3=(\frac{-b}{2})^3$.}
 
 \section{Jung´s method for power series of multiplicity three }
 
@@ -585,8 +588,8 @@ N^\Delta_i& 25& {\bf 26}& {\bf 52}& {\bf 78}& {\bf 104}& {\bf 130}& {\bf 156}& {
 \hline
 \hline
 \nu_i&2&3&5&7&9&11&13&6&11\\
-\chi()&-11&1&-1&0&0&0&-2&1&-1\\
-E_i^2& -13& -2& -4& -2& -2& -2& -1&-2&-1\\
+\chi()&-13&1&-1&0&0&0&-2&1&-1\\
+E_i^2& -11& -2& -4& -2& -2& -2& -1&-2&-1\\
 \hline
 \end{array}$$
 
@@ -967,8 +970,8 @@ In this case, the Newton polyhedra of $F$ contains a unique two dimensional comp
 
 
 \item If $M_1^3 = M_2^2$, the point corresponding to the monomial $M_1z$ lies in the unique dimensional compact face of the Newton polyhedron of $F$. The face equation is equal to 
-$$z^3 + c_1z + c_2.$$
-Therefore, $F$ is non degenerate with respect to its Newton polyhedron if and only if $3(c_1)^2 \ne 2(c_2)^3$.
+$$z^3 -3 c_1z + 2 c_2.$$
+Therefore, $F$ is non degenerate with respect to its Newton polyhedron if and only if \textcolor{blue}{$(c_1)^3 \ne (c_2)^2$}.
 
 
 \begin{tikzpicture}
@@ -1137,7 +1140,7 @@ $$J_{\check \tau_0}=\frac{2}{\nu_1(3m_2s + 2\nu_1+m_2)}$$
  We remark that this case it is equivalent to $f=z(z^2 - x^{m_2})$.
 
  
- \subsubsection{Third case $f= z^3 - x^{m_1}z^2 - x^{m_3}$ and $\omega = x^{\nu_1-1}dxdz$ such that $m_3 > m_1$} 
+ \subsubsection{Third case $f= z^3 - x^{m_1}z^2 - x^{m_3}$ and $\omega = x^{\nu_1-1}dxdz$ such that $m_3 >  3 m_1$} 
    Newton polyhedra has five compact faces: the 0-dimensional faces $\tau_0$ corresponding to the monomial $z^3$, $\tau_1$ corresponding to the monomial $x^{m_1}z^2$, and $\tau_2$ corresponding to the monomial $ x^{m_3}$, and the 1-dimensional faces  $\sigma_1$ contained in the hyperplanes
    $$p + m_1r = 3m_1$$
   and $\sigma_2$ contained in the hyperplanes
@@ -1179,9 +1182,9 @@ $$J_{\check \tau_0}=\frac{m_1}{\nu_1(3m_1s + \nu_1+m_1)}$$
  
  
  Similarly we have 
-  $$\zeta_f (t) = $$
- 
- We remark that this case is equivalent to $f=(z-x^{m_1})(z^2 - x^{m_3-m_2})$
+  $$\zeta_f (t) =  \frac{(t^{3m_1}-1)(t^{m_3}-1) }{(t-1)(t^3-1)} $$
+   $$\zeta_f (t) = \frac{(t^{3m_1}-1)(t^{2m_3}-1) }{(t-1)(t^3-1)(t^{m_3}-1)}$$
+ We remark that this case is equivalent to $f=(z-x^{m_1})(z^2 - x^{m_3-m_1})$
  
 
   \subsection{Trinomial forms for strata $E^\circ_{i,j}$}  
@@ -1286,7 +1289,7 @@ $$Z(F_9, \omega_9)_{o_9}=Z(F_{9,31}, \omega_{9,31})_{o_{9,31}}= Z(F_{17}, \omega
  $$f_7=f_{15}=z^3 - x^{78}, \quad f_{7,33}=f_{15,34}=z^3 - x^{52}z + x^{78}y,$$
  and $$\omega_7=\omega_{15}=\omega_{7,33}=\omega_{15,34}= x^{13}yzdxdydz,$$
 From the formulas given in the previous section;  we have 
-$$Z(F_7, \omega_7)_{o_7}=Z(F_{7,33}, \omega_{7,33})_{o_{7,33}}= Z(F_{15}, \omega_{15})_{o_{15}}=Z(F_{15,34}, \omega_{15,34})_{o_{15,34}} = \frac{1}{26(s^2 +3/2 s + 1/2}$$
+$$Z(F_7, \omega_7)_{o_7}=Z(F_{7,33}, \omega_{7,33})_{o_{7,33}}= Z(F_{15}, \omega_{15})_{o_{15}}=Z(F_{15,34}, \omega_{15,34})_{o_{15,34}} = \frac{1}{26(s^2 +3/2 s + 1/2}.$$
 \end{example}
 
   \subsection{Second reduction: Recursive elimination of exceptional  $(-1)$-curve not intersecting the strict transform of the discrimiant}
@@ -1296,6 +1299,9 @@ $$Z(F_7, \omega_7)_{o_7}=Z(F_{7,33}, \omega_{7,33})_{o_{7,33}}= Z(F_{15}, \omega
     \begin{proposition}\label{red2} 
     \end{proposition}
     
+    \begin{example} The Proposition \ref{red2} can be applied twice to eliminate the curves $E_9$ and $E_8$. First, $E_9$ is an exceptional  $(-1)$-curve that does not intersect the strict transform of the discriminant, and $E_8$ is an exceptional with $E_8^2=-2$ After blowing down $E_9$, the exceptional curve $E_8$ becomes a $(-1)$-curve. It does not intersect the strict transform of the discriminant. 
+    \end{example}
+    
     
   \subsection{Third reduction: $\QQ$-resolution of the discriminant}
  

From 9a6fc444b753c8359ba398c2a85fddf802fbc4e5 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Edwin=20Le=C3=B3n?= <eleon@unizar.es>
Date: Fri, 20 Sep 2024 16:45:58 +0200
Subject: [PATCH 4/4] Nuevo Tex

---
 TeX/Germs_Multiplicity_3_V0.pdf |  Bin 0 -> 350709 bytes
 TeX/Germs_Multiplicity_3_V0.tex | 1535 +++++++++++++++++++++++++++++++
 deformaciones3Edwin.ipynb       |  517 +++++++++++
 3 files changed, 2052 insertions(+)
 create mode 100644 TeX/Germs_Multiplicity_3_V0.pdf
 create mode 100644 TeX/Germs_Multiplicity_3_V0.tex
 create mode 100644 deformaciones3Edwin.ipynb

diff --git a/TeX/Germs_Multiplicity_3_V0.pdf b/TeX/Germs_Multiplicity_3_V0.pdf
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+%-------------------------------------------------------
+
+\begin{document}
+	
+	\title{On germs of hypersurfaces with multiplicity 3}
+	
+	\author{EAB, MGV, ELC}
+	
+	\maketitle
+	
+	
+	Consider a the germ of a multiplicity 3 singular point of hypersurface. An equation for such a germ it is  given by a monic Weierstrass polynomial $F = z^3 + a_1z^2 + a_2z + a_3 \in \CC[[{\bf x}]][z]$ of degree 3 with respect to $z$. Assume that $D_z F$ are not zero, and that $F$ is degenerate with respect to its Newton polyhedron. If it is non degenerate, the partial resolution associated with the dual of the Newton polyhedron of $F$ gives plenty information of the geometry and topology of the singular point of the hypersurface defined by $F$. 
+	
+	We propose the following general method.
+	
+	\medskip
+	
+	\noindent \emph{First step:} By means of a Tschirnhaus transformation, reduce $F$ to a monic Weierstrass equation $F = z^3 -3 b_2z + 2 b_3$ that does not have terms of degree 2 with respect to $z$. Assume that  the coefficients of degree one and zero with respect to $z$ are both non zero, i.e., the product $b_2b_3 \ne 0$.  If either $b_2 =0$ and $b_3 \ne 0$ or $b_2 \ne 0$ and $b_3=0$ it reduces to the case of a suspension that we have studied elsewhere. See \cite{}. 
+	
+	%We propose the following general method to study germs of multiplicity 3 singular points of hypersurfaces. Firstly, consider an equation of the germ given by a monic Weierstrass polynomial $F = z^3 + a_1z^2 + a_2z + a_3 \in \CC[[{\bf x}]][z]$ of degree 3 with respect to $z$. Assume that $D_z F$ is not zero. We will be mainly interested in the case when $F$ is degenerate with respect to its Newton polyhedron. Otherwise, the partial resolution associated with the dual of the Newton polyhedron of $F$ gives plenty information of the geometry and topology of the singular point of the hypersurface defined by $F$. By means of a Tschirnhaus transformation, reduce $F$ to a monic Weierstrass equation $z^3 + b_2z + b_3$ that does not have terms of degree 2 with respect to $z$. Assume that  the coefficients of degree one and zero with respect to $z$ are both non zero, i.e., the product $b_2b_3 \ne 0$.  
+	
+	\medskip
+	
+	\noindent \emph{Second step:} Consider an embedded resolution $\pi: X \rightarrow \CC^2$ of the reduced divisor given by the product $G=b_2b_3D_zF$. Consider the usual stratification of the pair $(X, (\pi \circ G)^{-1} \{G=0\})$ defined in terms of decomposition $\cup_{j \in J} E_j$ of $(\pi \circ G)^{-1} \{G=0\}$ into irreducible components. Pulling back $F$ under the map $\pi \times id$, we obtain for each stratum $E_I^\circ$ a \emph{quasi-trinomial local expression} for $F$. Section \ref{3nomial} clasifies the 5 posible quasi-trinomial local forms having terms of degrees 3, 1, and 0 with respect to $z$. These are the quasi-trinomials forms that appear at this step because we have previously made a Tschirnhaus transformation. Notice that the local expression is quasi-trinomial due to the fact that $\pi$ resolves both $b_2$ and $b_3$. Moreover, the quasi-trinomial local form is quasi-ordinary because $\pi$ also resolves the discriminant $D_zF$. The local quasi-trinomial expression will be non degenerate with respect to its Newton polyhedron unless the monomials $M_1$, $M_2$ and their coefficients $c_1$ and $c_2$ satisfy a precise condition. {\color{red}In principle, this precise condition can be defined on a subset of a stratum of $E_I^\circ$, with $\emptyset \ne I \subset J$. If it is the case, we consider the corresponding refinement of the stratification.} Moreover, the degeneracies of the degenerate trinomial local case are either on horizontal faces or of the form $(z-af(x,y))(z-bf(x,y))$ with $a, b \in \CC^\ast$, and $f \in \CC[x,y]$.  
+	
+	\medskip
+	
+	\noindent \emph{Third step:} Make translations of the type $z \mapsto z + h(x,y)$, where $h(x,y)$ is determined by the local quasi-ordinary expansion, to obtain a new set of coordinates such that the local form is non degenerate. Notice that after the translation terms of degree 2 respect to $z$ appear.  In fact, we obtain a new type of quasi-trinomial local form with terms of degree $3$, $2$, and $0$ with respect to $z$.
+	
+	\medskip
+	
+	\noindent \emph{Fourth step:} Use the stratification principle with the stratification introduced in the Second Step, the quasi-trinomial local forms and, if needed, the corresponding local differential forms to compute \emph{additive / multiplicative local invariants} of the singular point defined by $f$ at $o$. For instance, we will interested in the A'Campo zeta function and in the topological zeta function.
+	
+	\medskip
+	
+	A couple of remarks are in order after these four steps. 
+	
+	First, these steps achieve a resolution of the meromorphic function defined by $\frac{b_2^3}{b_3^2}$ and therefore the irreducible components $E_j$  of $(\pi \circ F)^{-1} \{F=0\}$ can be classified into 5 different types according to the image of $e_j$ under the meromorphic function $\mathcal{M}$ defined by  $\frac{b_2^3}{b_3^2}$. There are components $E_j$ that are mapped to $0$ called zeros, components $E_j$ that are mapped to $\infty$ called poles, components $E_j$ that are mapped surjectively onto $\mathbb{P}^1$ called dicritical, components that are mapped to the value 1, and finally components that are mapped to other values different  than 0, 1, and $\infty$. The image of a component $E_j$ is determined by the orders of vanishing of  $f$, $g$ and $\mathcal{M}= \frac{b_2^3}{b_3^2}$ along the divisor $E_j$ and (in some cases by) their pattern of intersection. 
+	
+	Second, the trinomial part of any quasi-trinomial local form, given by the formal restriction of the quasi-trinomial form to its Newton polyhedron, is determined by the valuations associated with $f$, $g$ and $\mathcal{M}= \frac{b_2^3}{b_3^2}$ too. 
+	
+	The following reductions are possible to perform the Fourth Step.
+	
+	\medskip
+	
+	\noindent \emph{Reduction A:} It is enough to consider the associated trinomial local forms instead of the quasi-trinomial local forms.
+	
+	\medskip
+	
+	\noindent \emph{Reduction B:} The strata produced by the strict transforms of $b_2$ and $b_3$ can be omitted, i.e., the contribution to the invariant due to a 0-dimensional stratum determined by an irreducible component of the strict transform of $b_2$ (resp. $b_3$)  equals the contribution of a single point of the 1-dimensional stratum to whose adherence the 0-dimensional stratum belongs.
+	
+	\medskip
+	
+	\noindent \emph{Reduction C:} Bamboos consisting of irreducible components that are mapped to $0$ (resp. to $\infty$ and containing a $(-1)$-curve can be omitted too. 
+	
+	\medskip
+	
+	The upshot of Reductions A, B, and C is that it is enough to consider the stratification associated to the minimal resolution $\pi' : X' \rightarrow \CC^2$ of $D_zF$ together with the trinomial local forms, and if needed the corresponding differential forms to compute the invariants of the singular point defined by $f$ at $0$.
+	
+	\medskip
+	
+	\noindent \emph{Reduction D:} It is enough to consider a minimal $\QQ$-resolution of the discriminant $D_ZF$.
+	
+	\medskip
+	
+	After all these reductions, we will prove that ...
+	
+	
+	\section{Power series of multiplicity three}
+	
+	
+	
+	Consider a the germ of a multiplicity 3 singular point of hypersurface. An equation for such a germ it is  given by a monic Weierstrass polynomial $F = z^3 + a_1z^2 + a_2z + a_3 \in \CC[[{\bf x}]][z]$ of degree 3 with respect to $z$. Assume that $D_z F$ is not zero, and that $F$ is degenerate with respect to its Newton polyhedron. 
+	
+	\begin{definition} The Tschirnhaus transformation associated to $F$ is the transformation given by $z \mapsto z- \frac{a_1}{3}$.
+	\end{definition}
+	
+	After the Tschirnhaus transformation, the monic Weierstrass polynomial $F(x,y,z - a_1/3)$ can be written as 
+	$$ z^3 -3b_2z + 2b_3$$
+	with
+	$$b_2 =  \frac{a_1^2}{9} - \frac{ a_2}{3} \quad {\rm and } \quad b_3= \frac{a_1^3}{27}-\frac{a_1a_2}{6} + \frac{a_3}{2}$$
+	and its the discriminant $D_zF$ of $F$ with respect to $F$ is of the form $108(b_2^3 - b_3^2)$.  
+	
+	
+	Denote the decompositions of $b_2$ and $b_3$ into irreducible components as
+	\begin{equation}
+		b_2 =  \prod_{i=1}^n h_i^{n_i} \prod_{j=1}^m k_j f  \qquad {\rm and} \qquad b_3 = \prod_{i=1}^n h_i \prod_{j=1}^m k_j^{m_j}g
+	\end{equation}
+	where $h_i, k_j$ are irreducible and $h_i, k_j, f, g$ are pairwise coprime and $n_i, m_j \geq 1$.
+	
+	
+	\begin{proposition}
+		The  discriminant $D_zF$ is of the form
+		$$ \prod_{i=1}^n h_i^{2} \prod_{j=1}^m k_j^3 \left (  \prod_{i=1}^n h_i^{3n_i-2} f^3 - \prod_{j=1}^m h_i^{2m_j -3} g^2 \right ).$$
+	\end{proposition}
+	
+	
+	\begin{proposition}
+		The singular point is isolated if and only if the disciminant $D_zF$ is reduced.
+	\end{proposition}
+	
+	
+	\begin{proposition}\label{prop.non-deg}
+		The monic Weierstrass polynomial $z^3 -3b_2z + 2b_3$ is non degenerate unless $M_1(c_1+A)$ and $b_3 = M_2(c_2 + B)$ with $M_1^3 = M_2^2$ and \textcolor{red}{$2c_1=3c_2$}
+	\end{proposition}
+	
+	
+	\section{Jung´s method for power series of multiplicity three }
+	
+	\subsection{Embbeded resolution of the discriminant and classification of their exceptional divisors}
+	
+	Consider an embedded resolution $\pi: X \rightarrow \CC^2$ of the reduced divisor given by the product $G=b_2b_3D_zF$. Consider the usual stratification of the pair $(X, (\pi \circ G)^{-1} \{G=0\})$ defined in terms of decomposition $\cup_{j \in J} E_j$ of $(\pi \circ G)^{-1} \{G=0\}$ into irreducible components, i.e., for each subset $I \subset J$ we define
+	$$E_I^\circ = \bigcap_{i \in I} E_j  \setminus  \bigcup_{j \not \in I} E_j .$$ 
+	
+	\begin{definition}For each irreducible component $E_j$ of of $(\pi \circ G)^{-1} \{G=0\}$ we define the numbers
+		$$N^0_j = {\rm ord}_{E_j} \pi^\ast b_2, \quad N^\infty_j = {\rm ord}_{E_j} \pi^\ast b_3, \quad  N^\Delta_j = {\rm ord}_{E_j} \pi^\ast D_zF,  \quad  {\rm and} \quad \nu_j =  {\rm ord}_{E_j} \pi^\ast \omega -1$$
+		where $\omega = dx \wedge dy$ is the standard volume form of $\CC^2$.
+	\end{definition}
+	
+	\begin{proposition} Let  $\pi': X' \rightarrow \CC^2$ be the minimal resolution of $D_zF$. Then, $\pi'$ is a resolution of the meromorphic function  $\mathcal{M} : \CC^2 \rightarrow \mathbb{P}^1$ defined by $\frac{b_2^3}{b_3^2}$ too.
+	\end{proposition} 
+	
+	There are four possibilities for the behaviour of the numbers $N^0_j, N^\infty_j$ and $N^\Delta_j$ associated with an irreducible component $E_j$ of $(\pi)^{-1} \{G=0\}$
+	\begin{itemize}
+		\item $3 N^0_j > 2N^\infty_j = N^\Delta_j$  and it is called a \emph{zero} because  in this case $\mathcal{M}(E_j)=0$;
+		\item  $2 N^\infty_j > 3N^0_j = N^\Delta_j$ and it is called a \emph{pole} because in this case  $\mathcal{M}(E_j)=\infty$;
+		\item $3 N^0_j = 2N^\infty_j = N^\Delta_j$ and in this case either it is called  \emph{dicritical}  because  $\mathcal{M}(E_i)=\mathbb{P}^1$  or it is mapped constantly to a value different than $0$, $1$ or $\infty$ under $\mathcal{M}$;
+		\item $3 N^0_j  = 2N^\infty_j  < N^\Delta_j$ and in this case it is mapped constantly to $1$ under $\mathcal{M}$.
+	\end{itemize}
+	
+	
+	\begin{remark} 
+		In the third case,  $N^\Delta_j$ is divisible by 6. In order to distinguish between dicritical divisors and divisors that are mapped constantly to a value  different than $0$, $1$ or $\infty$, one notices that connected components of dicritical divisors must intersect both to zeros and poles and connected components of divisors mapped constantly to a value  different than $0$, $1$ or $\infty$ do not intersect zeros or poles but do intersect dicritical divisors.
+		
+		
+		If the strata $E^\circ_{i,j} \ne \emptyset$ and $\mathcal{M}(E_i)=1$ then either $\mathcal{M}(E_j)=1$ or $E_j$ is dicritical, i.e., $\mathcal{M}(E_j)=\mathbb{P}^1$. 
+	\end{remark}
+	
+	
+	
+	\begin{definition}\label{extriplepole} For a divisor of th last type, i.e., a divisor that is mapped constantly to $1$, the difference
+		$N^\Delta_j - 3 N^0_j = N^\Delta_j - 2 N^\infty_j$ is called the \emph{excess of the discriminant} and it is denoted by $M^\Delta_j$.
+	\end{definition}
+	
+	
+	
+	\begin{example} Consider $F=z^3 - 3b_2z + 2b_3$ with 
+		$b_2= ((y^2 + x^3)^2 + x^5y)((x^2 + y^3)^2 + y^5x)$ 
+		and 
+		$b_3= (y^3+x^4)(x^3+y^4)$. The minimal resolution $\pi$ of the product $G=b_2 b_3 D_zF$ is obtained after 30 blow-ups.  The following figure and table give the dual resolution graph of $G$ and its associated numerical data.  
+		
+		
+		
+		
+		\begin{figure}[ht]
+			\begin{center}
+				\begin{tikzpicture}[vertice/.style={draw,circle,fill,minimum size=0.2cm,inner sep=0}]
+					\coordinate (U) at (0,0);
+					\node[vertice] at (U) {};
+					\node[below] at (U) {{\small $E_1$}};
+					\coordinate (G1) at (-0.5,0.75);
+					\coordinate (G2) at (-0.6,0.75);
+					\coordinate (G3) at (-0.4,0.75);
+					\coordinate (G4) at (-0.3,0.75);
+					\coordinate (G5) at (-0.1,0.75);
+					\coordinate (G6) at (-0.2,0.75);
+					\coordinate (G7) at (0,0.75);
+					\node[above] at (G7) {{\small ${{F^\Delta_i}}_{{}_{i=16}}^{{}{28}}$}};
+					\coordinate (G8) at (0.1,0.75);
+					\coordinate (G9) at (0.2,0.75);
+					\coordinate (G10) at (0.3,0.75);
+					\coordinate (G11) at (0.4,0.75);
+					\coordinate (G12) at (0.5,0.75);
+					\coordinate (G13) at (0.6,0.75);
+					
+					
+					
+					\coordinate (A1) at (1,0);
+					\coordinate (A2) at (2,0);
+					\coordinate (A3) at (3,0);
+					\coordinate (A4) at (4,0);
+					\coordinate (A5) at (5,0);
+					\coordinate (A6) at (6,0);
+					\coordinate (A7) at (7,0);
+					\coordinate (A8) at (5,1);
+					
+					
+					
+					\coordinate (B1) at (-1,0);
+					\coordinate (B2) at (-2,0);
+					\coordinate (B3) at (-3,0);
+					\coordinate (B4) at (-4,0);
+					\coordinate (B5) at (-5,0);
+					\coordinate (B6) at (-6,0);
+					\coordinate (B7) at (-7,0);
+					\coordinate (B8) at (-5,1);
+					
+					
+					\coordinate (F1) at (1.2,0.75);
+					\node[above] at (F1) {{\small $F^{\infty}_{33}$}};
+					\coordinate (F2) at (1.95,0.75);
+					\node[above] at (F2) {{\small $F^\Delta_{29}$}};
+					\coordinate (F3) at (6,0.75);
+					\node[above] at (F3) {{\small $F^{0}_{31}$}};
+					\coordinate (F4) at (-1.2,0.75);
+					\node[above] at (F4) {{\small $F^{\infty}_{34}$}};
+					\coordinate (F5) at (-1.95,0.75);
+					\node[above] at (F5) {{\small $F^\Delta_{30}$}};
+					\coordinate (F6) at (-6,0.75);
+					\node[above] at (F6) {{\small $F^{0}_{32}$}};
+					
+					
+					
+					
+					\node[vertice] at (A1) {};
+					\node[below] at (A1) {{\small $E_7$}};
+					\node[vertice] at (B1) {};
+					\node[below] at (B1) {{\small $E_{15}$}};
+					\node[vertice] at (A2) {};
+					\node[below] at (A2) {{\small $E_6$}};
+					\node[vertice] at (A3) {};
+					\node[below] at (A3) {{\small $E_5$}};
+					\node[vertice] at (A4) {};
+					\node[below] at (A4) {{\small $E_4$}};
+					\node[vertice] at (A5) {};
+					\node[below] at (A5) {{\small $E_3$}};
+					\node[vertice] at (A6) {};
+					\node[below] at (A6) {{\small $E_9$}};
+					\node[vertice] at (A7) {};
+					\node[below] at (A7) {{\small $E_8$}};
+					\node[vertice] at (A8) {};
+					\node[above] at (A8) {{\small $E_2$}};
+					
+					
+					\node[vertice] at (B2) {};
+					\node[below] at (B2) {{\small $E_{14}$}};
+					\node[vertice] at (B3) {};
+					\node[below] at (B3) {{\small $E_{13}$}};
+					\node[vertice] at (B4) {};
+					\node[below] at (B4) {{\small $E_{12}$}};
+					\node[vertice] at (B5) {};
+					\node[below] at (B5) {{\small $E_{11}$}};
+					\node[vertice] at (B6) {};
+					\node[below] at (B6) {{\small $E_{17}$}};
+					\node[vertice] at (B7) {};
+					\node[below] at (B7) {{\small $E_{16}$}};
+					\node[vertice] at (B8) {};
+					\node[above] at (B8) {{\small $E_{10}$}};
+					
+					\draw (U)--(A1)--(A2)--(A3)--(A4)--(A5)--(A6)--(A7);
+					\draw (A5)--(A8);
+					\draw (B5)--(B8);
+					\draw (U)--(B1)--(B2)--(B4)--(B4)--(B5)--(B6)--(B7);
+					
+					
+					\draw[->] (A1)--(F1);
+					\draw[->] (A1)--(F2);
+					\draw[->] (A6)--(F3);
+					\draw[->] (U)--(G1);
+					\draw[->] (U)--(G2);
+					\draw[->] (U)--(G3);
+					\draw[->] (U)--(G4);
+					\draw[->] (U)--(G5);
+					\draw[->] (U)--(G6);
+					\draw[->] (U)--(G7);
+					\draw[->] (U)--(G8);
+					\draw[->] (U)--(G9);
+					\draw[->] (U)--(G10);
+					\draw[->] (U)--(G11);
+					\draw[->] (U)--(G12);
+					\draw[->] (U)--(G13);
+					
+					
+					\draw[->] (B1)--(F4);
+					\draw[->] (B1)--(F5);
+					\draw[->] (B6)--(F6);
+				\end{tikzpicture}
+			\end{center}
+		\end{figure}
+		
+		
+		$$\begin{array}{|c||c|c|c|c|c|c|c|c|c|c|}
+			\hline 
+			&E_1&E_2, E_{10}&E_3, E_{11}&E_4, E_{12}&E_5, E_{13}&E_6, E_{14}&E_7, E_{15} &E_8, E_{16}&E_9, E_{17}\\
+			\hline
+			\hline
+			N^f_i& {\bf 8}& 10& 20& 28& 36& 44& {\bf 52}& 21&42\\
+			N^g_i& {\bf12}& {\bf 13}& {\bf 26}& {\bf 39}& {\bf 52}& {\bf 65}& {\bf 78}& {\bf 26}& {\bf 52}\\
+			N^\Delta_i& 25& {\bf 26}& {\bf 52}& {\bf 78}& {\bf 104}& {\bf 130}& {\bf 156}& {\bf 52}&{\bf 104}\\
+			\hline
+			\hline
+			\nu_i&2&3&5&7&9&11&13&6&11\\
+			\chi()&-11&1&-1&0&0&0&-2&1&-1\\
+			E_i^2& -13& -2& -4& -2& -2& -2& -1&-2&-1\\
+			\hline
+		\end{array}$$
+		
+		
+		
+		The divisor $E_1$ is mapped to the value 1, the divisors $E_7$ and $E_{15}$ are dicriticals, and the rest of exceptionals divisors are zeros. The 13 arrows $F^\Delta_i$ with $i=16, \dots, 29$ attached to the divisor $E_1$ correspond to irreducible components of the strict transforms of $D_zF$, the arrows attached to the divisor $E_7$ (resp. $E_{13}$) correspond to an irreducible component  strict transform of $b_3$ and another irreducible component of $D_zF$. Finally the arrow attched to $E_9$ (resp. to $E_{17}$) correspond to the strict transform of $b_2$.
+		%$$\pi^{-1} \{\mathcal{M} = 0\}  = 2 (F^{0}_1 + F^{0}_2) + 4 (E_2 + E_5 + E_{10} + E_{13})  + 6 (E_4 + E_{12} )  + 8 (E_3 + E_{11})  + 11 (E_8+E_{16}) + 22 (E_9 + E_{17}))  , \quad \pi^{-1} \{\mathcal{M} = 1\} = \sum_{i=1}^{15} F^\Delta_i + E_1$$
+		$$\pi^{-1} \{\mathcal{M} = 0\}  = 2  ( \sum_{31,32}F^{0}_i  + \sum_{6,14} E_i ) + 4 \sum_{2,5,10,13}E_i  + 6 \sum_{4,12 }E_i   + 8 \sum_{3,11}E_i   + 11\sum_{8,16} E_i  + 22 \sum_{9,17} E_i ,$$ 
+		$$\pi^{-1} \{\mathcal{M} = \infty\} = 3 F^{\infty}_{33} + 3 F^{\infty}_{34}, \quad   \pi(E_7) = \pi(E_{15}) = \mathbb{P}^1, \quad \pi^{-1} \{\mathcal{M} = 1\} = \sum_{i=16}^{30} F^\Delta_i + E_1$$
+		
+		
+		
+		The stratification associated with the resolution $\pi$ consist of $35$  strata $E^\circ_{i,j}$ of dimension 0 and 11 strata $E_i^\circ$ of dimension 1 and $\chi(E_i^\circ) \ne 0$. 
+		The former correspond to the elements $(i,j)$ of the set 
+		$$S= \left \{ \begin{array}{ccccccccc}
+			(1,7)& (1,15)& (1,16)& (1,17)& (1,18)& (1,19)& (1,20)& (1,21)& (1,22)\\
+			(1,23)& (1,24)& (1, 25)& (1, 26)& (1, 27)& (1, 28)&  (2,3)& (3,4)& (3,9)\\
+			(4,5)& (5,6)& (6,7)& (7, 29)& (7, 33)& (8, 9)& (9,31)& (10,11)& (11,12)\\
+			(11,17)& (12, 13)& (13,14)& (14,15)& (15,30)& (15, 34)& 16,17)& (17,32)
+		\end{array} \right \} $$
+		%the 13 intersections $E_1 \cap F_i^\Delta$ for $i=16, \dots,  28$ and the 6 intersections 
+		%$$  E_7 \cap F_{34}^\infty, \quad   E_7 \cap F_{30}^\Delta,  \quad E_9 \cap F_{32}^0, \quad E_{15} \cap F_{35}^\infty,   \quad E_{15} \cap F_{31}^\Delta,  \quad E_{17} \cap F_{33}^0 $$
+		and the latter correspond with the indexes
+		$$1, \quad 2, \quad  3, \quad  7, \quad  8, \quad  9, \quad 10, \quad  11, \quad  15,  \quad  16, \quad  19.$$
+		
+		Notice that the minimal resolution $\pi'$ of the discriminant $D_zF$ is obtained after just 11 blow-ups. The blowups corresponding to divisors $E_8, E_9, E_{14}$ and $E_{15}$ are needed to resolve $b_2$.
+	\end{example}
+	
+	
+	\subsection{Trinomial forms associated to the strata} 
+	
+	Let us consider the birational map $\tilde{\pi} := ( \pi, id ):  \tilde{X} : =X \times \CC \rightarrow \CC^{3}$, where $id: \CC \rightarrow \CC$ is the identity map, and the pullback of $F$ under $\pi \times id$. The pullback $\tilde{\pi}^\ast F$ can be expressed in terms of the pullbacks of $b_2$ and $b_3$ as
+	\begin{equation}
+		z^3 -3 \pi^\ast b_2 z +  2 \pi^\ast b_3.
+	\end{equation}
+	over each non empty strata $E^\circ_I$ of $X$ created by the resolution $\pi$. 
+	Since $\pi$ is a resolution of both $b_2$ and $b_3$, over each non empty strata $E^\circ_I$ of $X$ created by the resolution $\pi$,  the terms $(\pi^\ast b_3)_{|_I}$ and $(\pi^\ast b_3)_{|_I}$ can be reexpressed in terms of the multiplicities $N^0_i$ and $N^\infty_i$ with $i \in I$ as 
+	\begin{equation}
+		(\pi^\ast b_2)_{|_I}=  x_1^{N^0_{i_1}} y_1^{N^0_{i_2}} u_1(x_1, y_1) \quad {\rm and } \quad 
+		(\pi^\ast b_3)_{|_I}=  x_1^{N^\infty_{i_1}} y_1^{N^\infty_{i_2}} u_1(x_1, y_1)
+	\end{equation}
+	where $u_1, u_2$  are units in $\CC\{x_1, y_1\}$. Hence, we have expressions 
+	\begin{equation}\label{quasi-trinomial}
+		z^3 -3 x_1^{N^0_{i_1}} y_1^{N^0_{i_2}} u_1(x_1, y_1) z +  2 x_1^{N^\infty_{i_1}} y_1^{N^\infty_{i_2}}  u_2(x_1,y_1),
+	\end{equation}
+	for each non empty strata $E^\circ_I$. Notice that these expressions are quasi-ordinary because $\pi$ is also a resolution of $D_zF$ and therefore $D_z(\pi^\ast F)_{|_I}$ is the product of a monomial in the variables $x_1$ and $y_1$ and a unit in $\CC\{x_1, y_1\}$. 
+	
+	According to Section \ref{3nomial} the expression (\ref{quasi-trinomial}) is non degenerate with respect to its Newton polyhedron except for the case described in Proposition \ref{prop.non-deg}, i.e., the case when the there is a divisor $E_i$ with $i \in I$ such that $\mathcal{M}(E_i)=1$ or equivalently $3N^0_i=2N^\infty_i < N^\Delta_i$. 
+	
+	\begin{remark} 
+		If the strata $E^\circ_{i,j} \ne \emptyset$ and $\mathcal{M}(E_i)=1$ then either $\mathcal{M}(E_j)=1$ or $E_j$ is dicritical, i.e., $\mathcal{M}(E_j)=\mathbb{P}^1$. 
+	\end{remark}
+	
+	
+	\begin{proposition} There exist a change of coordinates of the form
+		$$x_1= x_2, \quad y_1= y_2, \quad z = z_1 - x_2^\frac{N^0_i}{2} y_2^\frac{N^0_j}{2} + \cdots$$
+		such that (\ref{quasi-trinomial}) becomes
+		either
+		$$z_1^3 + x_2^{\frac{N^0_i}{2}} y_2^{\frac{N^0_j}{2}} u'_1(x_2,y_2) z_1^2   + x_2^{N_i^\infty + M_i^\Delta} y_2^{N_j^\infty + M_j^\Delta} u'_2(x_2,y_2)$$
+		if both  $\mathcal{M}(E_i) =  \mathcal{M}(E_j)=1$
+		or
+		$$z_1^3 + x_2^{\frac{N^0_i}{2}}y_2^{\frac{N^0_j}{2}} u'_1(x_2,y_2) z_1^2   + x_2^{N_i^\infty + M_i^\Delta}y_2^{N_j^\infty } u'_2(x_2,y_2)$$
+		if  $\mathcal{M}(E_i)=1$ and  $E_j$ is dicritical
+	\end{proposition}
+	
+	\begin{definition}The trinomial form  $f_I$ associated with the stratum $E^\circ_I$ is 
+		$$\begin{cases}
+			z^3 - x^{N^\infty_{i}}, & \mathcal{M}(E_i) \ne 1, \infty \\
+			z^3 - x^{N^0_{i}}  z +  x^{N^\infty_{i}}, &  \mathcal{M}(E_i) = \infty \\
+			z^3 - x_2^{\frac{N^0_i}{2}}z^2  + x^{N_i^\infty + M_i^\Delta} , &   \mathcal{M}(E_i) = 1, \\
+		\end{cases}$$
+		in the case $I= \{i\}$ 
+		and 
+		$$\begin{cases}
+			z^3 + x^{N^\infty_i}y^{N^\infty_j}, & \mathcal{M}(E_i), \mathcal{M}(E_j) \ne 1, \infty \\
+			z^3 + x^{\frac{N^0_i}{2}}y^{\frac{N^0_j}{2}}z^2  + x^{N_i^\infty + M_i^\Delta}y^{N_j^\infty + M_j^\Delta} , &  \mathcal{M}(E_i) =  \mathcal{M}(E_j) = 1 \\
+			z^3 + x^{\frac{N^0_i}{2}}y^{\frac{N^0_j}{2}} z^2 + x^{N_i^\infty + M_i^\Delta}y^{N_j^\infty} , &  \mathcal{M}(E_i) = 1,  \mathcal{M}(E_j) = \mathbb{P}^1 \\
+			z^3 + x^{N^0_{i}}y^{N^0_{j}}  z + x^{N^\infty_{i}}y^{N^\infty_{j}}, & {\rm else}\\
+		\end{cases}$$
+		in the case $I= \{i, j\}$. 
+	\end{definition}
+	
+	
+	\begin{example} The following tables show all the trinomial forms and differential forms for the all the 0-dimensional strata and all the 1-dimensional strata with not zero Euler characteristic of Example \ref{extriplepole}
+		
+		
+		$$\begin{array}{c|c|c|c}
+			i&f_i&\omega_i& \chi()\\
+			\hline
+			1& z^3 + x^4z^2 + x^{13}& (2, 1, 1)&  -13\\
+			2,10& z^3 + x^{10}z + x^{13}& (3, 1, 1)&1\\
+			3,11& z^3 + x^{20}z + x^{26}& (5, 1, 1)& -1\\
+			7,15& z^3 + \color{red}{x^{52}z} + x^{78}& (13, 1, 1)& -2\\
+			8, 16&  z^3 + x^{21}z + x^{26}& (6, 1, 1)& 1\\
+			9,17& z^3 + x^{42}z + x^{52}& (11, 1, 1)& -1\\
+		\end{array}$$
+		
+		
+		
+		$$\begin{array}{c|c|c}
+			(i,j)&f_{i,j}&\omega_{i,j}\\
+			\hline
+			&&\\
+			(1, 7), (1, 15)& z^3 + x^4y^{26}z^2 + x^{13}y^{78}& (2, 13, 1)\\
+			(2,3),  (10, 11)& z^3 + x^{10}y^{20}z + x^{13}y^{26}& (3, 5, 1)\\
+			(3, 9), (11, 17)& z^3 + x^{20}y^{42}z + x^{26}y^{52}& (5, 11, 1)\\
+			(3, 4), (11,12)& z^3 + x^{20}y^{28}z + x^{26}y^{39}& (5, 7, 1)\\
+			(8, 9), (16,17)& z^3 + x^{21}y^{42}z + x^{26}y^{52}& (6, 11, 1)\\
+			(4, 5), (12,13)& z^3 + x^{28}y^{36}z + x^{39}y^{52}& (7, 9, 1)\\
+			(5, 6), (13,14)& z^3 + x^{36}y^{44}z + x^{52}y^{65}& (9, 11, 1)\\
+			(6, 7), (14,15)& z^3 + x^{44}y^{52}z + x^{65}y^{78}& (11, 13, 1)\\
+			(1, i), i=16, \dots, 28&  z^3 + x^4z^2 + x^{13}y& (2, 1, 1)\\
+			(9, 31), (17, 32)& z^3 + x^{42}yz + x^{52}& (11, 1, 1)\\
+			(7, 33), (15, 34)& z^3 + x^{52}z + x^{78}y& (13, 1, 1)\\
+			(7, 29) (15, 30)& z^3 + x^{26}z^2 + x^{78}y& (13, 1, 1)\\
+			%(12, 'f')& z^3 + x^{42}yz + x^{52}& (11, 1, 1)\\
+			%(16, 'g')& z^3 + x^{52}z + x^{78}y& (13, 1, 1)\\
+			%(16, 'd')& z^3 + x^{26}z^2 + x^{78}y&  (13, 1, 1)
+		\end{array}$$
+		
+		
+	\end{example}
+	
+	
+	
+	
+	\section{The quasi-trinomial case}\label{3nomial}
+	In this section we consider the case of 
+	$$F = z^3 + M_1(c_1+A)z + M_2(c_2 + B) \in \CC[[{\bf x}]][z]$$
+	where $M_1={\bf x}^{m_1}$ and $M_2={\bf x}^{m_2}$ are monomials, $c_1,c_2 \in \CC^\ast$ and $A, B \in ({\bf x})\CC[[{\bf x}]]$
+	
+	We distinguish the following two cases
+	\begin{itemize}
+		
+		\item Assume that the monomials $M_1$ and $M_2$ are not related by the divisibility relation, i.e., neither $M_1 \nmid M_2$ nor $M_2 \nmid M_1$. 
+		In this case, the Newton polyhedra of $F$ contains a unique two dimensional compact face determined by the monomials $z^3$, $M_1z$ and $M_2$. Moreover, $F$ is non degenerate with respect to its Newton polyhedra.
+		
+		\
+		
+		
+		\begin{tikzpicture}
+			\fill [yellow] (1,0) rectangle (3,-2);
+			\fill [yellow] (-2,0) rectangle (1,2);
+			\draw[gray, thick] (-2,-2) -- (-2,2);
+			\draw[gray, thick] (-2,-2) -- (3,-2);
+			\draw[gray, thick] (-2,0) -- (3,0);
+			\draw[gray, thick] (1,-2) -- (1,2);
+			\filldraw[black] (-2,-2) circle (2pt) node[anchor=west]{};
+			\filldraw[black] (1,0) circle (2pt) node[anchor=south west]{$m_2$};
+			\filldraw[black] (2,-1) circle (2pt) node[anchor=south west]{$m_1$};
+			
+			
+			
+			
+			
+			
+			
+			
+			
+			
+			
+			
+			\fill [yellow] (11,0,4)--(11,0,0)--(13,0,0)--(13,0,4)--(11,0,4);
+			\fill [yellow] (11,0,4)--(11,0,7)--(6,0,7)--(6,0,4)--(11,0,4);
+			
+			\draw[gray, thick] (6,0,0) -- (6,0,7);
+			\draw[gray, thick] (6,0,0) -- (6,4,0);
+			\draw[gray, thick] (6,0,0) -- (13,0,0);
+			
+			
+			
+			
+			
+			\filldraw[black] (6,0,0) circle (2pt) node[anchor=west]{};
+			
+			
+			\draw[gray, thick] (6,0,4) -- (13,0,4);
+			\draw[gray, thick] (11,0,0) -- (11,0,7);
+			
+			
+			
+			\draw[gray, thick] (6,3,0) -- (11,0,4);
+			\draw[gray, thick] (9,1,6) -- (11,0,4);
+			\draw[gray, thick] (6,3,0) -- (9,1,6);
+			
+			
+			\draw[gray, dashed] (9,0,0) -- (9,0,6);
+			\draw[gray, dashed] (6,0,6) -- (9,0,6);
+			\draw[gray, dashed] (9,1,0) -- (9,1,6);
+			\draw[gray, dashed] (6,1,6) -- (9,1,6);
+			\draw[gray, dashed] (9,0,6) -- (9,1,6);
+			
+			\draw[gray, dashed] (6,1,0) -- (6,1,6);
+			\draw[gray, dashed] (6,1,0) -- (9,1,0);
+			\draw[gray, dashed] (6,0,6) -- (6,1,6);
+			\draw[gray, dashed] (9,0,0) -- (9,1,0);
+			
+			
+			\filldraw[black] (6,3,0) circle (2pt) node[anchor=south west]{$z^3$};
+			\filldraw[black] (11,0,4) circle (2pt) node[anchor=north west]{$M_2$};
+			\filldraw[black] (9,0,6) circle (2pt) node[anchor=north]{$M_1$};
+			\filldraw[black] (9,1,6) circle (2pt) node[anchor=south west]{$M_1z$};
+		\end{tikzpicture}
+		
+		\
+		
+		
+		
+		\item Assume that the monomials $M_1$ and $M_2$  are related by the divisibility relation, i.e., either $M_1 | M_2$ or $M_1 | M_2$. In this case the Newton polyhedron of $F$ is a polygonal path. We distinguish the following cases. 
+		
+		\
+		
+		
+		\begin{tikzpicture}
+			\fill [yellow] (6,-2) rectangle (9,0);
+			\fill [yellow] (9,0) rectangle (11,2);
+			\draw[gray, thick] (6,-2) -- (6,2);
+			\draw[gray, thick] (6,-2) -- (11,-2);
+			\draw[gray, thick] (6,0) -- (11,0);
+			\draw[gray, thick] (9,-2) -- (9,2);
+			\filldraw[black] (6,-2) circle (2pt) node[anchor=west]{};
+			\filldraw[black] (9,0) circle (2pt) node[anchor=north west]{$m_2$};
+			\filldraw[black] (9.5, 1.3) circle (2pt) node[anchor=north west]{$m_1$};
+		\end{tikzpicture}
+		
+		\
+		
+		
+		\begin{enumerate}
+			\item If the point $3m_1$ does not belong to  the set $2 m_2 + (\RR_{\geq 0})^2$, the Newton polyhedron of $F$ consists of two segments. The first one is determined by the monomials $z^3$ and $M_1z$ and the second is determined by the monomials $M_1z$ and $M_2$. it follows that $F$ is non degenerate with respect to its newton polyhedron.
+			
+			
+			
+			\
+			
+			
+			\begin{tikzpicture}
+				\fill [yellow] (-2,-2) rectangle (1,-2/3);
+				\fill [yellow] (-2,-2) rectangle (0,0);
+				\draw[gray, thick] (-2,-2) -- (-2,2);
+				\draw[gray, thick] (-2,-2) -- (3,-2);
+				\draw[gray, thick] (-2,0) -- (3,0);
+				\draw[gray, thick] (1,-2) -- (1,2);
+				
+				\draw[gray, dashed] (0,-2/3) -- (0,0);
+				\draw[gray, dashed] (0,-2/3) -- (1,-2/3);
+				
+				
+				
+				\filldraw[black] (-2,-2) circle (2pt) node[anchor=west]{};
+				\filldraw[black] (1,0) circle (2pt) node[anchor=north west]{$m_2$};
+				\filldraw[black] (0, -2/3) circle (2pt) node[anchor=north west]{$\frac{2}{3}m_2$};
+				\filldraw[black] (-0.5, -0.2) circle (2pt) node[anchor=north east]{$m_1$};
+				
+				
+				
+				
+				\fill [yellow] (28/3,0,4)--(28/3,0,8/3)--(11,0,8/3)--(11,0,0)--(6,0,0)--(6,0,4)--(28/3,0,4);
+				
+				
+				\draw[gray, thick] (6,0,0) -- (6,0,7);
+				\draw[gray, thick] (6,0,0) -- (6,4,0);
+				\draw[gray, thick] (6,0,0) -- (13,0,0);
+				
+				
+				
+				
+				
+				\filldraw[black] (6,0,0) circle (2pt) node[anchor=west]{};
+				
+				
+				\draw[gray, thick] (6,0,4) -- (13,0,4);
+				\draw[gray, thick] (11,0,0) -- (11,0,7);
+				
+				
+				
+				\draw[gray, thick] (8,1,3) -- (11,0,4);
+				\draw[gray, thick] (6,3,0) -- (8,1,3);
+				
+				
+				\draw[gray, dashed] (8,1,3) -- (8,0,3);
+				\draw[gray, dashed] (8,0,0) -- (8,0,3);
+				\draw[gray, dashed] (8,1,3) -- (8,1,0);
+				\draw[gray, dashed] (8,1,0) -- (8,0,0);
+				\draw[gray, dashed] (6,1,0) -- (8,1,0);
+				
+				
+				\draw[gray, dashed] (6,1,0) -- (6,1,3);
+				\draw[gray, dashed] (6,1,3) -- (8,1,3);
+				\draw[gray, dashed] (6,1,3) -- (6,0,3);
+				\draw[gray, dashed] (6,0,3) -- (8,0,3);
+				
+				
+				
+				
+				\draw[gray, dashed] (28/3,0,8/3) -- (28/3,0,4);
+				\draw[gray, dashed] (28/3,0,8/3) -- (11,0,8/3);
+				
+				\filldraw[black] (6,3,0) circle (2pt) node[anchor=south west]{$z^3$};
+				\filldraw[black] (11,0,4) circle (2pt) node[anchor=north west]{$M_2$};
+				\filldraw[black] (8,0,3) circle (2pt) node[anchor=north]{$M_1$};
+				\filldraw[black] (8,1,3) circle (2pt) node[anchor=south west]{$M_1z$};
+				
+				
+				
+				
+			\end{tikzpicture}
+			
+			\
+			
+			
+			
+			\item If the point $3m_1$ belongs to  the set $2 m_2 + (\RR_{\geq 0})^2$, the Newton polyhedron of $F$ consists of one segment with vertex corresponding to the monomials $z^3$ and $M_2$. We distinguish to further cases.
+			
+			
+			
+			
+			
+			\begin{enumerate}
+				\item If $M_1^3 \ne M_2^2$, the point corresponding to the monomial $M_1z$ is an interior point of the Newton polyhedron of $F$. It follows that $F$ is non degenerate with respect to its Newton polyhedron.
+				
+				
+				
+				\begin{tikzpicture}
+					
+					
+					\fill [yellow] (28/3,0,4)--(28/3,0,8/3)--(11,0,8/3)--(11,0,0)--(6,0,0)--(6,0,4)--(28/3,0,4);
+					
+					
+					\draw[gray, thick] (6,0,0) -- (6,0,7);
+					\draw[gray, thick] (6,0,0) -- (6,4,0);
+					\draw[gray, thick] (6,0,0) -- (13,0,0);
+					
+					\filldraw[black] (6,0,0) circle (2pt) node[anchor=west]{};
+					
+					
+					\draw[gray, thick] (6,0,4) -- (13,0,4);
+					\draw[gray, thick] (11,0,0) -- (11,0,7);
+					
+					
+					\draw[gray, thick] (6,3,0) -- (11,0,4);
+					
+					\draw[gray, thick] (6,3,0) -- (10.5,1,3) -- (12.75, 0, 4.5);
+					
+					
+					\draw[gray, thick] (6,3,0) -- (6,3,7);
+					\draw[gray, thick] (6,3,0) -- (13,3,0);
+					
+					
+					\draw[gray, dashed] (6,1,3) -- (6,0,3);
+					\draw[gray, dashed] (10.5,0,0) -- (10.5,0,3);
+					\draw[gray, dashed] (6,1,3) -- (6,1,0);
+					\draw[gray, dashed] (10.5,1,0) -- (10.5,0,0);
+					\draw[gray, dashed] (6,1,0) -- (10.5,1,0);
+					
+					
+					\draw[gray, dashed] (10.5,1,0) -- (10.5,1,3);
+					\draw[gray, dashed] (10.5,1,3) -- (6,1,3);
+					\draw[gray, dashed] (10.5,1,3) -- (10.5,0,3);
+					\draw[gray, dashed] (10.5,0,3) -- (6,0,3);
+					
+					
+					
+					
+					\draw[gray, dashed] (28/3,0,8/3) -- (28/3,0,4);
+					\draw[gray, dashed] (28/3,0,8/3) -- (11,0,8/3);
+					
+					\filldraw[black] (6,3,0) circle (2pt) node[anchor=south west]{$z^3$};
+					\filldraw[black] (11,0,4) circle (2pt) node[anchor=north west]{$M_2$};
+					\filldraw[black] (10.5,0,3) circle (2pt) node[anchor=north]{};
+					\filldraw[black] (10.5,1,3) circle (2pt) node[anchor=south west]{$M_1z$};
+					\filldraw[black] (12.75,0,4.5) circle (2pt) node[anchor=north]{};
+					
+					
+					
+				\end{tikzpicture}
+				
+				
+				
+				\item If $M_1^3 = M_2^2$, the point corresponding to the monomial $M_1z$ lies in the unique dimensional compact face of the Newton polyhedron of $F$. The face equation is equal to 
+				$$z^3 + c_1z + c_2.$$
+				Therefore, $F$ is non degenerate with respect to its Newton polyhedron if and only if $3(c_1)^2 \ne 2(c_2)^3$.
+				
+				
+				\begin{tikzpicture}
+					
+					
+					\fill [yellow] (28/3,0,4)--(28/3,0,8/3)--(11,0,8/3)--(11,0,0)--(6,0,0)--(6,0,4)--(28/3,0,4);
+					
+					
+					\draw[gray, thick] (6,0,0) -- (6,0,7);
+					\draw[gray, thick] (6,0,0) -- (6,4,0);
+					\draw[gray, thick] (6,0,0) -- (13,0,0);
+					
+					\filldraw[black] (6,0,0) circle (2pt) node[anchor=west]{};
+					
+					
+					\draw[gray, thick] (6,0,4) -- (13,0,4);
+					\draw[gray, thick] (11,0,0) -- (11,0,7);
+					
+					
+					\draw[gray, thick] (6,3,0) -- (11,0,4);
+					
+					
+					
+					\draw[gray, thick] (6,3,0) -- (6,3,7);
+					\draw[gray, thick] (6,3,0) -- (13,3,0);
+					
+					
+					\draw[gray, dashed] (6,1,8/3) -- (6,0,8/3);
+					\draw[gray, dashed] (28/3,0,0) -- (28/3,0,8/3);
+					\draw[gray, dashed] (6,1,8/3) -- (6,1,0);
+					\draw[gray, dashed] (28/3,1,0) -- (28/3,0,0);
+					\draw[gray, dashed] (6,1,0) -- (28/3,1,0);
+					
+					
+					\draw[gray, dashed] (28/3,1,0) -- (28/3,1,8/3);
+					\draw[gray, dashed] (28/3,1,8/3) -- (6,1,8/3);
+					\draw[gray, dashed] (28/3,1,8/3) -- (28/3,0,8/3);
+					\draw[gray, dashed] (28/3,0,8/3) -- (6,0,8/3);
+					
+					
+					
+					
+					\draw[gray, dashed] (28/3,0,8/3) -- (28/3,0,4);
+					\draw[gray, dashed] (28/3,0,8/3) -- (11,0,8/3);
+					
+					\filldraw[black] (6,3,0) circle (2pt) node[anchor=south west]{$z^3$};
+					\filldraw[black] (11,0,4) circle (2pt) node[anchor=north west]{$M_2$};
+					\filldraw[black] (28/3,0,8/3) circle (2pt) node[anchor=north]{};
+					\filldraw[black] (28/3,1,8/3) circle (2pt) node[anchor=south west]{$M_1z$};
+					
+					
+					
+					
+				\end{tikzpicture}
+				
+				
+			\end{enumerate}
+		\end{enumerate}
+		
+		
+		
+		
+	\end{itemize}
+	
+	
+	\newpage
+	
+	\section{Contibutions to the topological zeta function and the A´Campo monodromy zeta function of the trinomial forms}
+	
+	
+	\begin{proposition}\label{formestrpi}
+		The monodromy zeta function $\zeta_F(t)$ and the local topological  zeta function $Z(F,s)_0$ of $F$ equal
+		$$(1 - t^3)^{\alpha} \prod_{I} \zeta_{f_I}(t)^{-\chi(E_I^\circ)}$$ 
+		$$\sum_I \chi(E^\circ_I) Z(f_I, \omega_I)_{o_I}$$
+		where $I$ varies over the statification associated to the resolution $\pi$ of $b_2b_3D_zF$, the exponent $\alpha$ is the number of points of blow-ups needed to obtain $\pi$, and $o_I$ is a point in $E^\circ_I$.
+	\end{proposition}
+	
+	\begin{proof} It follows from the the formula for $\zeta_f$ in terms of a partial resolution \cite[Theorem 1]{GSLM97} and the  Stratum  principle for topological zeta functions \cite[Stratum Principle 1.2]{ABCNLM02}. Each of the $\alpha$  factors of the form   $(1-t^3)$ of $\zeta_f$ comes from the intersection of Milnor fiber of $F$ at $o$ and the line thru a center of blow-up of $\pi$. Each of these intersection sets consists of 3 points that are cyclically permuted by the monodromy. 
+	\end{proof}
+	
+	
+	\begin{example} The resolution $\pi$ of Example \ref{extriplepole} consist of $\alpha=15$ blow-ups and the associated stratification has 35 strata. According to Proposition \ref{formestrpi} we have
+		$$\zeta_F(t) = \frac{(1-t^3)^{15} \zeta^{}_{f_{2}} \zeta^{2}_{f_{4}} \zeta^{2}_{f_{5}} \zeta^{2}_{f_{6}}  \zeta^{}_{f_{8}} \zeta^{}_{f_{10}} \zeta^{2}_{f_{12}} \zeta^{2}_{f_{13}} \zeta^{2}_{f_{14}}  \zeta^{}_{f_{16}}  }
+		{\zeta^{13}_{f_{1}} \zeta^{}_{f_{3}} \zeta^{4}_{f_{7}} \zeta^{}_{f_{9}} \zeta^{}_{f_{11}} \zeta^{4}_{f_{15}} \zeta^{}_{f_{17}} }= \Phi^3_1 \Phi^{15}_2 \Phi_3 \Phi^2_6 \Phi^2_{13} \Phi^{15}_{26} \Phi^2_{78}$$
+		where $\Phi_i$ is $i$-th the cyclotomic polymonial, and $Z(F,s)_0$ equals 
+		$$-13Z(f_1,s)_0 + \sum_{i=2,8,10,16}Z(f_i,s)_0 -  \sum_{i=3, 9,11,17} Z(f_i,s)_0 - 2 \sum_{i=7,15}Z(f_i,s)_0 + \sum_{(i,j) \in S} Z(f_{i,j},s)_0=$$
+		$$\frac{-1/26s^3 + 2749/6084s^2 + 1969/4056s + 1/8}{(s+1)(s+1/2)^3}$$
+		where $S$ is the subset indexing the 0-dimensional strata of Example \ref{extriplepole} and we have omitted the symbols of differential forms associated to each strata for shortness sake. 
+	\end{example} 
+	
+	
+	
+	\subsection{Trinomial forms for strata $E^\circ_i$}  
+	
+	\subsubsection{First case $f= z^3 - x^{m_3}$ and $\omega = x^{\nu_1-1}dxdz$}
+	Newton polyhedra has three compact faces: the  0-dimensional faces $\tau_0$ corresponding to the monomial $z^3$ and $\tau_1$ corresponding to the monomial $x^{m_3}$, and the 1-dimensional face $\sigma$ contained in 
+	$$ 3p + m_3r = 3m_3$$
+	The corresponding candidate pole is
+	$$ 3m_3 s + 3\nu_1 +m_3$$
+	
+	The topological zeta function  $Z_{{\rm top}}(f, \omega,s)_0$ equals
+	%a*c^2*s - m*s - 3*a*s - m - 3*a
+	$$ \frac{3\nu_1 + m_3 + (3\nu_1 + m_3 - \nu_1 (m_3, 3)^2)s}{\nu_1(3m_3 s + 3\nu_1 + m_3)(s+1)}$$
+	or equivalently   $J_{\check \tau_0}-\frac{(3, m_3)J_{\check \sigma}s}{s+1}+J_{\check \tau_1}$  with the terms $J_{\check \tau_0}$, $J_{\check \tau_1}$ and $J_{\check \sigma}$  given as follows.
+	\begin{enumerate}
+		\item The dual cone $\check \tau_1$ to the face $\tau_1$ is simplicial and generated  by the rays  $$\frac{1}{(m_3,3)} (3, m_3), e_2.$$ This cone has multiplicity $\frac{3}{(m_3,3)}$. Its contribution to the local topological zeta function is
+		$$J_{\check \tau_1}= \frac{3}{ \nu_1 (3m_3 s + 3\nu_1 + m_3)}.$$
+		
+		
+		\item The dual cone $\check \sigma$ of the face $\sigma$ is  the simplical cone generated by the ray $$\frac{1}{(m_3,3)} (3,  m_3).$$  Its contribution to the local zeta function is
+		$$J_{\check \sigma}= \frac{(3,m_3)}{(3m_3 s + 3\nu_1 + m_3)}.$$
+		
+		
+		\item The dual cone $\check \tau_0$ to the face  $\tau_0$ is the cone generated by the rays $$\frac{1}{(m_3,3)} (3, m_3), e_1.$$ This cone has multiplicity $\frac{m_3}{(m_3,3)}$.  Its contribution to the the local topological zeta function is
+		$$J_{\check \tau_0}= \frac{m_3}{(3m_3 s + 3\nu_1 + m_3)}.$$
+	\end{enumerate}
+	Similarly we have 
+	$$\zeta_f (t) = \frac{(1-t^3)(1-t^{m_3})}{(1-t^{3m_3})^{(m_3,3)}}$$
+	
+	
+	
+	
+	\subsubsection{Second case $f= z^3 - x^{m_2}z - x^{m_3}$ and $\omega = x^{\nu_1-1}dxdz$ such that $3m_2 < 2 m_3$.}
+	Newton polyhedra has five compact faces: the 0-dimensional faces $\tau_0$ corresponding to the monomial $z^3$, $\tau_1$ corresponding to the monomial $x^{m_2}z$, and $\tau_2$ corresponding to the monomial $ x^{m_3}$, and the 1-dimensional faces  $\sigma_1$ contained in the hyperplane
+	$$2p + m_2r = 3m_2$$
+	and $\sigma_2$ contained in the hyperplane
+	$$p + (m_3-m_2)r = m_3$$
+	The candidate poles are 
+	$$3m_2s + 2 \nu_1 + m_2 \quad {\rm and } \quad  m_3s + \nu_1 + m_3-m_2$$
+	
+	The topological zeta function $Z_{{\rm top}}(f, \omega,s)_0$ equals 
+	$$\frac{(2-\nu_1(m_2,2)^2)s + 2(\nu_1 + 1)}{(3m_2s + 2 \nu_1 + m_2)(s+1)}$$  
+	or equivalently   $J_{\check \tau_0}-(m_2,2)J_{\check \sigma_1}\frac{s}{s+1}+J_{\check \tau_1}  - \frac{J_{\check \sigma_2}s}{s+1}+J_{\check \tau_2}$  with the terms $J_{\check \tau_0}$, $J_{\check \tau_1}$, $J_{\check \tau_2}$, $J_{\check \sigma_1}$ and $J_{\check \sigma_2}$  given as follows. Note that the candidate pole $ m_3s + \nu_1 + m_3-m_2$ is a false candidate pole.
+	
+	\begin{enumerate}
+		\item The dual cone $\check \tau_2$ to the face $\tau_2$ is the regular cone generated by the rays 
+		$$(1, m_3 - m_2), e_2$$
+		give the contribution
+		$$J_{\check \tau_2}=\frac{1}{(m_3s + \nu_1 + m_3-m_2)}$$
+		
+		\item The dual cone $\check \sigma_2$ of the face $\sigma_2$ is  the regular cone  generated by the rays
+		$$(1, m_3 - m_2)$$
+		give the contribution
+		$$J_{\check \sigma_2} = \frac{1}{(m_3s + \nu_1 + m_3-m_2)}$$
+		
+		
+		\item The dual cone $\check \tau_1$ to the face $\tau_1$ is the simplicial cone generated by the rays 
+		$$(1,  m_3 - m_2), \frac{1}{(m_2,2)}(2, m_2)$$
+		The multiplicity of this cone is $\frac{m_2+2m_3}{(2,m_2)}$.  Its contribution to the the local topological zeta function is
+		$$J_{\check \tau_1} =  \frac{3m_3-2m_2}{(m_3s + \nu_1 + m_3-m_2)(3m_2s + 2 \nu_1 + m_2)}$$
+		
+		\item The dual cone $\check \sigma_1$ of the face $\sigma_1$ is  the cone  generated by the rays
+		$$ \frac{1}{(n_2,2)} (2,  m_2)$$
+		It gives the contribution
+		$$J_{\check \sigma_1} =  \frac{(m_2, 2)}{(3m_2s + 2\nu_1 + m_2)}$$
+		
+		\item The dual cone $\check \tau_0$ of the face $\tau_0$ is the simplicial cone generated by the rays 
+		$$e_1,  \frac{1}{(m_2,2)}(2, m_2)$$
+		The multiplicity of this cone is $\frac{2}{(2,m_2)}$.   Its contribution to the the local topological zeta function is
+		$$J_{\check \tau_0}=\frac{2}{\nu_1(3m_2s + 2\nu_1+m_2)}$$
+	\end{enumerate}
+	Similarly we have 
+	$$\zeta_f (t) = \frac{(1-t^3)}{(1-t^{3m_2})^{(m_2,2)}}$$
+	
+	
+	We remark that this case it is equivalent to $f=z(z^2 - x^{m_2})$.
+	
+	
+	\subsubsection{Third case $f= z^3 - x^{m_1}z^2 - x^{m_3}$ and $\omega = x^{\nu_1-1}dxdz$ such that $m_3 > m_1$} 
+	Newton polyhedra has five compact faces: the 0-dimensional faces $\tau_0$ corresponding to the monomial $z^3$, $\tau_1$ corresponding to the monomial $x^{m_1}z^2$, and $\tau_2$ corresponding to the monomial $ x^{m_3}$, and the 1-dimensional faces  $\sigma_1$ contained in the hyperplanes
+	$$p + m_1r = 3m_1$$
+	and $\sigma_2$ contained in the hyperplanes
+	$$2p + (m_3-m_1)r = 2m_3$$
+	The candidate poles are 
+	$$3m_1s +  \nu_1 + m_1 \quad {\rm  and } \quad  2m_3s + 2\nu_1 + m_3-m_1$$
+	
+	The topological zeta function $Z_{{\rm top}}(f, \omega,s)_0$ equals 
+	$$\frac{ (2m_1m_3 + 6m_1\nu_1 - 2m_3\nu_1 -3m_1\nu_1c^2)s^2    + ( 3m_1m_3  - m_1^2  + 8m_1\nu_1  - \nu_1^2c^2s  - m_1\nu_1c^2)s
+		- m_1^2 + m_1m_3 + m_1\nu_1 + m_3\nu_1 + 2\nu_1^2}{(s+1)(3m_1s +  \nu_1 + m_1)(2m_3s + 2\nu_1 + m_3-m_1)}$$  
+	or equivalently   $V_0-\frac{As}{s+1}+V_1  - \frac{A_2s}{s+1}+V_2$  with the terms $V_0$, $V_1$ and $A$  given as follows.
+	\begin{enumerate}
+		\item The dual cone $\check \tau_2$ to the face $\tau_2$ is the regular cone generated by the rays 
+		$$e_2, \frac{1}{(m_3-m_1,2)}(2, m_3-m_1)$$
+		give the contribution
+		$$J_{\check \tau_2}=\frac{2}{(2m_3s + 2\nu_1 + m_3-m_1)}$$
+		
+		\item The dual cone $\check \sigma_2$ of the face $\sigma_2$ is  the regular cone  generated by the rays
+		$$\frac{1}{(m_3-m_1,2)}(2, m_3-m_1)$$
+		give the contribution
+		$$J_{\check \sigma_2} = \frac{(m_3-m_1,2)}{(2m_3s + 2\nu_1 + m_3-m_1)}$$
+		
+		
+		\item The dual cone $\check \tau_1$ to the face $\tau_1$ is the simplicial cone generated by the rays 
+		$$ \frac{1}{(m_3-m_1,2)}(2, m_3-m_1), (1, m_1)$$
+		The multiplicity of this cone is $\frac{m_2+2m_3}{(2,m_2)}$.  Its contribution to the the local topological zeta function is
+		$$J_{\check \tau_1} =  \frac{m_3-3m_1}{(2m_3s + 2\nu_1 + m_3-m_1)(3m_1s +  \nu_1 + m_1)}$$
+		
+		\item The dual cone $\check \sigma_1$ of the face $\sigma_1$ is  the cone  generated by the ray
+		$$ (1,  m_1)$$
+		It gives the contribution
+		$$J_{\check \sigma_1} =  \frac{1}{(3m_1s + \nu_1 + m_1)}$$
+		
+		\item The dual cone $\check \tau_0$ of the face $\tau_0$ is the simplicial cone generated by the ray
+		$$e_1,  (1, m_1)$$
+		The multiplicity of this cone is $\frac{2}{(2,m_2)}$.   Its contribution to the the local topological zeta function is
+		$$J_{\check \tau_0}=\frac{m_1}{\nu_1(3m_1s + \nu_1+m_1)}$$
+	\end{enumerate}
+	
+	
+	Similarly we have 
+	$$\zeta_f (t) = $$
+	
+	We remark that this case is equivalent to $f=(z-x^{m_1})(z^2 - x^{m_3-m_2})$
+	
+	
+	\subsection{Trinomial forms for strata $E^\circ_{i,j}$}  
+	
+	\subsubsection{ First case  $f= z^3 - x^{m_3}y^{n_3}$ and $\omega = x^{\nu_1-1}y^{\nu_2-1}dxdydz$}
+	
+	Newton polyhedra has three compact faces: the  0-dimensional faces $\tau_0$ corresponding to the monomial $z^3$ and $\tau_1$ corresponding to the monomial $x^{m_3}y^{n_3}$, and the 1-dimensional face $\sigma$ contained in 
+	$$ 3p + m_3r = 3m_3, \qquad  3q + n_3r = 3n_3$$
+	The corresponding candidate poles are
+	$$ 3m_3 s + 3\nu_1 +m_3, \qquad 3n_3 s + 3\nu_2 + n_3$$
+	
+	The topological zeta function  $Z_{{\rm top}}(f, \omega,s)_0$ equals
+	$$ \frac{((3s+1)m_3n_3 + 3(\nu_1n_3+\nu_2m_3+3\nu_1\nu_2))(s+1) -3\nu_1\nu_2(3, m_3, n_3)^2s}{\nu_1\nu_2(3m_3 s + 3\nu_1 + m_3)(3n_3 s + 3\nu_2 + n_3)(s+1)}$$
+	or equivalently   $V_0-\frac{(3, m_3, n_3)As}{s+1}+V_1 $  with the terms $V_0$, $V_1$ and $A$  given as follows.
+	\begin{enumerate}
+		\item The dual cone $\check \tau_1$ to the face $\tau_1$ is simplicial and generated  by the rays $$\frac{1}{(m_3,3)} (3, 0, m_3), \frac{1}{(n_3,3)}  (0, 3, n_3) , e_3.$$ This cone has multiplicity $\frac{9}{(m_3,3)(n_3,3)}$. Its contribution to the local topological zeta function is
+		$$V_1= \frac{9}{ (3m_3 s + 3\nu_1 + m_3)(3n_3 s + 3\nu_2+n_3)}.$$
+		
+		
+		\item The dual cone $\check \sigma$ of the face $\sigma$ is  the simplical cone generated by the rays $$\frac{1}{(m_3,3)} (3, 0, m_3), \frac{1}{(n_3,3)}  (0, 3, n_3) .$$ This cone has multiplicity  $\frac{3 (m_3, n_3, 3)}{(m_3,3)(n_3,3)}$. Its contribution to the local zeta function is
+		$$A= \frac{3(m_3,n_3,3)}{(3m_3 s + 3\nu_1 + m_3)(3n_3 s + 3\nu_2+n_3)}.$$
+		
+		
+		\item The dual cone $\check \tau_0$ to the face  $\tau_0$ is the cone generated by the rays $$\frac{1}{(m_3,3)} (3, 0, m_3), \frac{1}{(n_3,3)}  (0, 3, n_3), e_1, e_2$$ which is not simplicial. Using the expression $\check \tau_0 = \mathbb{R}^3_{\geq 0} - \check \tau_1 - \check \sigma$,  its contribution to the the local topological zeta function is
+		$$V_0=\frac{1}{ \nu_1 \nu_2 (3s+1)} - \frac{9}{(3m_3 s + 3\nu_1 + m_3)(3n_3 s + 3\nu_2 + n_3)(3s+1)}$$
+		$$  = \frac{m_1m_2(3s+1) + 3(\nu_1m_2+\nu_2m_1)}{\nu_1\nu_2(3m_3 s + 3\nu_1 + m_3)(3n_3 s + 3\nu_2 + n_3)}$$
+	\end{enumerate}
+	
+	
+	%Comparing the contributions of the pairs $(f, \omega)$ and the case $(\bar{f}, \bar{\omega})$ with $\bar{f} = z^3 - \bar{x}^{\frac{m_3}{g}} \bar{y}^{\frac{n_3}{g}}$ and $\bar{\omega} = \bar{x}^{\frac{\nu_1}{g}-1} \bar{y}^{\frac{\nu_2}{g}-1} d \bar{x}d \bar{y} dz$
+	%we get the relations 
+	%$$\bar{V}_0= g V_0 \qquad \bar{V}_1= g V_1 \qquad \bar{A} = 
+	%\begin{cases}
+	%g A &  \hbox{ if either } 3 \mid m \hbox{ and } 3 \mid \frac{m}{g} \hbox{ or }  3 \nmid m \hbox{ and } 3 \nmid \frac{m}{g} \\
+	%\frac{g}{9} A & \hbox{ if } 3 \mid m \hbox{ and } 3 \nmid \frac{m}{g}
+	%\end{cases}$$
+	
+	
+	
+	
+	
+	\subsection{}  Second case $f= z^3 - x^{m_2}y^{n_2}z - x^{m_3}y^{n_3}$ and such that $m_2 \leq 2 m_3, 3 n_2 \leq 2 n_3$ and $3(m_2, n_2) \ne  2(m_3, n_3)$.
+	
+	Newton polyhedra has five compact faces: the 0-dimensional faces $\tau_0$ corresponding to the monomial $z^3$, $\tau_1$ corresponding to the monomial $x^{m_2}y^{n_2}z$, and $\tau_2$ corresponding to the monomial $ x^{m_3}y^{n_3}$, and the 1-dimensional faces  $\sigma_1$ contained in the hyperplanes
+	$$2p + m_2r = 3m_2, \qquad  2q + n_2r = 3n_2$$
+	and $\sigma_2$ contained in the hyperplanes
+	$$p + (m_3-m_2)r = m_3, \qquad q + (n_3-n_2)r =n_3$$
+	The candidate poles are 
+	$$3m_2s + 2 \nu_1 + m_2, \qquad 3n_2 s + 2\nu_2 + n_2$$
+	and 
+	$$m_3s + \nu_1 + m_3-m_2, \qquad n_3s + \nu_2 + n_3-n_2$$
+	
+	The topological zeta function is the sum of the following five terms 
+	\begin{enumerate}
+		\item The dual cone $\check \tau_2$ to the face $\tau_2$ is the regular cone generated by the rays 
+		$$(1, 0, m_3 - m_2), (0, 1 , n_3 - n_2), e_3$$
+		give the contribution
+		$$\frac{1}{(m_3s + \nu_1 + m_3-m_2)(n_3s + \nu_2 + n_3-n_2)}$$
+		
+		\item The dual cone $\check \sigma_2$ of the face $\sigma_2$ is  the regular cone  generated by the rays
+		$$(1, 0, m_3 - m_2), (0, 1 , n_3 - n_2)$$
+		give the contribution
+		$$\frac{-s}{(m_3s + \nu_1 + m_3-m_2)(n_3s + \nu_2 + n_3-n_2)(s+1)}$$
+		
+		
+		\item The dual cone $\check \tau_1$ to the face $\tau_1$ is the cone generated by the rays 
+		$$(1, 0, m_3 - m_2), (0, 1 , n_3 - n_2), \frac{1}{(m_2,2)}(2, 0, m_2), \frac{1}{(n_2,2)}(0,2,n_2)$$
+		which is not simplicial. Denote by $\rho$ the cone generated by $\frac{1}{(m_2,2)}(2, 0, m_2)$, $\frac{1}{(n_2,2)}(0,2,n_2)$  and $e_3$. Using the expression $\check \tau_1 =  \rho - \check \tau_2 - \check \sigma_2$,  its contribution to the the local topological zeta function is
+		$$\frac{4}{(m_2,2)(n_2,2)(3m_2s + 2\nu_1 + m_2)(3n_2s + 2\nu_2 + n_2)} - \frac{1}{(m_3s + \nu_1 + m_3-m_2)(n_3s + \nu_2 + n_3-n_2)}$$
+		
+		\item The dual cone $\check \sigma_1$ of the face $\sigma_1$ is  the cone  generated by the rays
+		$$ \frac{1}{(n_2,2)} (2, 0, m_2),  \frac{1}{(n_2,2)} (0, 2 , n_2)$$
+		This cone has multiplicity $2(m_2, n_2, 2)$. It gives the contribution
+		$$\frac{-2(m_2, n_2, 2)s}{(3m_2s + 2\nu_1 + m_2)(3n_2s + 2\nu_2 + n_2)(s+1)}$$
+		
+		\item The dual cone $\check \tau_0$ of the face $\tau_0$ is the cone generated by the rays 
+		$$e_1, e_2, \frac{1}{(m_2,2)}(2, 0, m_2), \frac{1}{(n_2,2)}(0,2,n_2)$$
+		Using the expression $\check \tau_0 =   \mathbb{R}^3_{\geq 0}  - \rho - \check \sigma_2$,  its contribution to the the local topological zeta function is
+		$$\frac{1}{\nu_1\nu_2(3s+1)} - \frac{2}{(3m_2s + 2\nu_1+m_2)(3n_2s + 2\nu_2+m_2)(3s+1)}$$
+	\end{enumerate}
+	
+	\subsection{} Third case $f= z^3 - x^{m_1}y^{n_1}z^2 - x^{m_3}y^{n_3}$ and such that $3m_1 \leq  m_3, 3 n_1 \leq  n_3$ and $3(m_1, n_1) \ne  2(m_3, n_3)$.
+	
+	\section{Reductions}
+	
+	\subsection{First reduction: Elimination of the 0-dimensional strata due to the strict transforms of $b_2$ and of $b_3$} The following Proposition shows that the o-dimensional strata associated to the strict transforms of $b_2$ and $b_3$ can eliminated. Each of these points can be added to the 1-dimensional strata satisfying that its clousure contains it, and the invariant is computed with the trinomial form of the 1-dimensional strata. 
+	
+	\begin{proposition}\label{red1}Assume that $E_i$ is an exceptional divisor of the resolution $\pi$ with $\pi(E_i)=0$ (resp. $\pi(E_i)=\infty$) and that  $F^0_j$ is an strict transform of $b_2$ (resp. $b_3$) such that $E_i \cap F_j \ne \emptyset$. Then
+		$$Z(F_i, \omega_i)_{o_i}=Z(F_{i,j}, \omega_{i,j})_{o_{i,j}} $$
+	\end{proposition}
+	
+	\begin{example} The proposition \ref{red1} can be applied to the strata 
+		$$E^\circ_{9,31}=E_9 \cap F^0_{31}, \quad E^\circ_{17,32}=E_{17} \cap F^0_{32}, \quad E^\circ_{7,33}= E_7 \cap F^\infty_{33} \quad \hbox{ and } \quad E^\circ_{15,34}=E_{15} \cap F^\infty_{34}$$
+		of Example \ref{extriplepole}. 
+		
+		For the strata due to the strict transforms $F^0_{31}$ and $F^0_{32}$ of $b_2$, note that $\mathcal{M}(E_9)= \mathcal{M}(E_{17})=0$. The corresponding  trinomial forms and differential forms are 
+		$$f_9=f_{17}=f_{9,31}=f_{17,32}=z^3 - x^{52}, \qquad  \omega_9=\omega_{17}=\omega_{9,31}=\omega_{17,32}= x^{11}yzdxdydz,$$
+		From the formulas given in the previous section;  we have 
+		$$Z(F_9, \omega_9)_{o_9}=Z(F_{9,31}, \omega_{9,31})_{o_{9,31}}= Z(F_{17}, \omega_{17})_{o_{15}}=Z(F_{17,32}, \omega_{17,32})_{o_{17,32}} = \frac{37/858s + 85/1716}{s^2 +241/156 s + 85/156}.$$
+		
+		For the strata due to the strict transforms $F^\infty_{33}$ and $F^\infty_{34}$ of $b_3$, note that $\mathcal{M}(E_7)= \mathcal{M}(E_{15})=0$. The corresponding  trinomial forms and differential forms are 
+		$$f_7=f_{15}=z^3 - x^{78}, \quad f_{7,33}=f_{15,34}=z^3 - x^{52}z + x^{78}y,$$
+		and $$\omega_7=\omega_{15}=\omega_{7,33}=\omega_{15,34}= x^{13}yzdxdydz,$$
+		From the formulas given in the previous section;  we have 
+		$$Z(F_7, \omega_7)_{o_7}=Z(F_{7,33}, \omega_{7,33})_{o_{7,33}}= Z(F_{15}, \omega_{15})_{o_{15}}=Z(F_{15,34}, \omega_{15,34})_{o_{15,34}} = \frac{1}{26(s^2 +3/2 s + 1/2}$$
+	\end{example}
+	
+	\subsection{Second reduction: Recursive elimination of exceptional  $(-1)$-curve not intersecting the strict transform of the discrimiant}
+	
+	After performing the First reduction, exceptional $(-1)$-curves that do not intersect the strict transform of the discriminant may appear. The next proposition shows that we can eliminante these divisors.. 
+	
+	\begin{proposition}\label{red2} 
+	\end{proposition}
+	
+	
+	\subsection{Third reduction: $\QQ$-resolution of the discriminant}
+	
+	\section{On generic projections}
+	
+	% In general, the $z$-axis is not a generic direction of projection for the embedded topological class of the surface given by the Weierstrass polynomial  $F=z^3-2f(x,y)z+3g(x,y)$. Therefore the discriminant of $F$ with respect to $z$ given by $f^2+g^3$ is not generic inside the same equivalence class. Let us describe a situation where it is possible to describe a generic topological deformation $F_t=z^3-2f_t(x,y)z+3g_t(x,y)$ of $F$ whose discriminant $f_t^3+g_t^2$ is generic too. 
+	
+	
+	%\begin{proposition} 
+	% Assume that the functions $f$ and $g$ admit factorizations of the form  $f= x^{2a} f_1$ and $g=x^{3a}g_1$. Moreover, assume that the the multiplicity of the polynomial $F_1(x, y, z)=  F(x, y, x^az_1)/x^{3a}$  at the origin is equal or less than $2$. Then, any pair of deformations $f_t$ and $g_t$ of the functions $f$ and $g$ gives a topological deformation $F_t$ of $F$ and a topological deformation  $f_t^3+g_t^2$ of $f_t^3+g_t^2$. Moreover, if $f_t^3+g_t^2$ is generic, then $F_t$ is generic.
+	%\end{proposition}
+	
+	\begin{example} Consider the Weierstrass polynomial  $F=z^3-3f(x,y)z+2g(x,y)$ with $f=y^2 - x^7 + x^5y$ and $g= f^2 + (y + x^2)^6$. It defines a germ of  isolated surface singularity with Milnor number $\mu=41$. The discriminant of $F$ is given by 
+		$$f^3-g^2 = f^3 - f^4 - 2f^2(y+x^2)^6 +(y+x^2)^{12}$$
+		and it has the topological type given by  three $(2,7)$-cusp with pairwise intersection multiplicity $16$. Its dual resolution graph is pictured  (left) bellow.
+		
+		\
+		
+		
+		\begin{tikzpicture}
+			\draw[gray, thick] (6,-2) -- (10,-2);
+			\draw[gray, thick] (9,-2) -- (9,0);
+			\filldraw[black] (9,-1) circle (2pt) node[anchor=west]{$E_6$};
+			\filldraw[black] (9,0) circle (2pt) node[anchor=west]{$E_7$};
+			\filldraw[black] (6,-2) circle (2pt) node[anchor=north]{$E_1$};
+			\filldraw[black] (7,-2) circle (2pt) node[anchor=north]{$E_2$};
+			\filldraw[black] (8,-2) circle (2pt) node[anchor=north]{$E_3$};
+			\filldraw[black] (9,-2) circle (2pt) node[anchor=north]{$E_5$};
+			\filldraw[black] (10,-2) circle (2pt) node[anchor=north]{$E_4$};
+			
+			\node (Z) at (9, 0) {};
+			\node (A) at (9, 1) {$d$};
+			\node (B) at (8, 1) {$d$};
+			\node (C) at (10, 1) {$d$};
+			\draw[->]  (Z) edge (A);
+			\draw[->]  (Z) edge (B);
+			\draw[->]  (Z) edge (C);
+			
+			
+			
+			
+			\draw[gray, thick] (16,-2) -- (20,-2);
+			\draw[gray, thick] (19,-2) -- (19,1);
+			\filldraw[black] (19,-1) circle (2pt) node[anchor=west]{$E_6$};
+			\filldraw[black] (19,0) circle (2pt) node[anchor=west]{$E_8$};
+			\filldraw[black] (19,1) circle (2pt) node[anchor=west]{$E_7$};
+			\filldraw[black] (16,-2) circle (2pt) node[anchor=north]{$E_1$};
+			\filldraw[black] (17,-2) circle (2pt) node[anchor=north]{$E_2$};
+			\filldraw[black] (18,-2) circle (2pt) node[anchor=north]{$E_3$};
+			\filldraw[black] (19,-2) circle (2pt) node[anchor=north]{$E_5$};
+			\filldraw[black] (20,-2) circle (2pt) node[anchor=north]{$E_4$};
+			
+			\node (W) at (19, -2) {};
+			\node (D) at (18, -1) {$d_t$};
+			\draw[->]  (W) edge (D);
+			\node (Y) at (19, 0) {};
+			\node (E) at (18, 0) {$d_t$};
+			\draw[->]  (Y) edge (E);
+		\end{tikzpicture}
+		
+		\
+		
+		The polinomial  $F_t=z^3 -3f_t(x,y)z - 2 g_t(x,y)$ with 
+		$$f_t= f  + t^2y^2 \qquad \hbox{ and } \qquad g_t = g + \frac{3}{2}tyf + t^3y^3$$ defines a $\mu$-constant topological deformation of $F$ whose discriminant $f_t^3-g_t^2$ has the following embedded topological type. The discriminant of $F_t$ has two branches. One is a $(2,7)$-cusp and the other branch has two Puiseux pairs, namely, $(2,7)$ and $(2,3)$. Their intersection multiplicity is $28$. The dual resolution graph of the discriminant is pictured (right) above. 
+		
+		The topological zeta function and the monodromy zeta function of $F$ and $F_t$ are given by 
+		$$ \frac{-41}{168} \frac{s^2 - 283/123s - 304/123}{(s + 16/21) (s + 19/24) (s + 1) } \quad \hbox{ and } \quad 
+		\Phi_1 \Phi_2 \Phi_3 \Phi_4 \Phi_6 \Phi_7 \Phi_8 \Phi_{12} \Phi_{21} \Phi_{24}$$
+		In particular, they both have  Milnor number equal to 41. This can be computed from embedded resolutions of the meromorphic functions $\mathcal{M}$ and $\mathcal{M}_t$ with the following numerical data 
+		$$\begin{array}{c|ccccccccc}
+			& E_1& E_2& E_3& E_4& E_5& E_6& E_7\\
+			\hline 
+			f& 2& 4& 6& 7& 14& 15& 16\\
+			g& 4& 8& 12& 12& 24& 24& 24\\
+			d& 6&12& 18& 21& 42& 45& 48
+		\end{array}
+		\qquad \begin{array}{c|ccccccccc}
+			& E_1& E_2& E_3& E_4& E_5& E_6& E_7& E_8\\
+			\hline 
+			f_t& 2& 4& 6& 7& 14& 14& 14& 28\\
+			g_t& 3& 6& 9& 11& 21& 21& 21& 42\\
+			d_t& 6&12& 18& 21& 42& 44& 45& 90
+		\end{array}$$
+		
+		\
+		
+		\noindent and dual resolution graphs
+		
+		\
+		
+		\begin{tikzpicture}
+			\draw[gray, thick] (6,-2) -- (10,-2);
+			\draw[gray, thick] (9,-2) -- (9,0);
+			\filldraw[black] (9,-1) circle (2pt) node[anchor=west]{$E_6$};
+			\filldraw[black] (9,0) circle (2pt) node[anchor=west]{$E_7$};
+			\filldraw[black] (6,-2) circle (2pt) node[anchor=north]{$E_1$};
+			\filldraw[black] (7,-2) circle (2pt) node[anchor=north]{$E_2$};
+			\filldraw[black] (8,-2) circle (2pt) node[anchor=north]{$E_3$};
+			\filldraw[black] (9,-2) circle (2pt) node[anchor=north]{$E_5$};
+			\filldraw[black] (10,-2) circle (2pt) node[anchor=north]{$E_4$};
+			
+			\node (Z) at (9, 0) {};
+			\node (A) at (9, 1) {$d$};
+			\node (B) at (8, 1) {$d$};
+			\node (C) at (10, 1) {$d$};
+			\node (F) at (8, 0) {$f$};
+			
+			\draw[->]  (Z) edge (A);
+			\draw[->]  (Z) edge (B);
+			\draw[->]  (Z) edge (C);
+			\draw[->]  (Z) edge (F);
+			
+			
+			\node (V) at (8, -2) {};
+			\node (G1) at (7.25, -1) {$g$};
+			\node (G2) at (7.5, -1) {$g$};
+			\node (G3) at (8.25, -1) {$g$};
+			\node (G4) at (8.5, -1) {$g$};
+			
+			
+			\draw[->]  (V) edge (G1);
+			\draw[->]  (V) edge (G2);
+			\draw[->]  (V) edge (G3);
+			\draw[->]  (V) edge (G4);
+			
+			
+			
+			\draw[gray, thick] (16,-2) -- (20,-2);
+			\draw[gray, thick] (19,-2) -- (19,1);
+			\filldraw[black] (19,-1) circle (2pt) node[anchor=west]{$E_6$};
+			\filldraw[black] (19,0) circle (2pt) node[anchor=west]{$E_8$};
+			\filldraw[black] (19,1) circle (2pt) node[anchor=west]{$E_7$};
+			\filldraw[black] (16,-2) circle (2pt) node[anchor=north]{$E_1$};
+			\filldraw[black] (17,-2) circle (2pt) node[anchor=north]{$E_2$};
+			\filldraw[black] (18,-2) circle (2pt) node[anchor=north]{$E_3$};
+			\filldraw[black] (19,-2) circle (2pt) node[anchor=north]{$E_5$};
+			\filldraw[black] (20,-2) circle (2pt) node[anchor=north]{$E_4$};
+			
+			\node (W) at (19, -2) {};
+			\node (D) at (18, -1) {$d_t$};
+			\draw[->]  (W) edge (D);
+			\node (Y) at (19, 0) {};
+			\node (E) at (18, 0) {$d_t$};
+			\draw[->]  (Y) edge (E);
+			
+			
+			\node (W) at (19, -2) {};
+			\node (F) at (17, -1) {$f_t$};
+			\node (G) at (20, -1) {$g_t$};
+			\draw[->]  (W) edge (F);
+			\draw[->]  (W) edge (G);
+			
+			
+			\node (V) at (20, -2) {};
+			\node (GG) at (21, -2) {$g_t$};
+			\draw[->]  (V) edge (GG);
+			
+		\end{tikzpicture}
+		
+		\
+		
+		
+		
+		
+		In order to that there check that both $F$ and $F_t$ define the same embedded topological type, notice that  
+		$$F_t(x,y,z+yt)=F(x,y,t) + 3tyz^2$$
+		Moreover, in the left-hand side change $z \mapsto z+ty$ is an isomorphism that does not change the discriminant with respect to $z$, nor the Milnor number. Finally, in the right-hand side we have a $\mu$-constant deformation of isolated hypersurface singularities of the form $f(x) + tg(x)$, that preserves the embedded topological type according to \cite{par:99}.
+		
+		
+		Finally, it is worth mentioning that the topological type of the discriminant of $F_t$ can be realizable as $f_1^3-g_1^2$ for another pair of functions $(f_1,g_1)$. The corresponding function $F_1= z^3-3f_1z + 2g_1$ does not belong to the embedded  topological type equivalence class of $F$ and $F_t$ because their Milnor number and monodromy zeta function do not coincide. The topological and monodromy zeta function of $F_1$ are 
+		$$ \frac{-5}{21} \frac{s - 16/5}{(s + 16/21)(s + 1)} \qquad \hbox{ and, } \qquad \Phi_1 \Phi_3 \Phi_7^2 \Phi_{21}^2 $$
+		In particular, the Milnor number of $F_1$ is $38$.
+		This can be computed from embedded resolution of the meromorphic functions $\mathcal{M}_1$ with the following numerical data 
+		$$\begin{array}{c|ccccccccc}
+			& E_1& E_2& E_3& E_4& E_5& E_6& E_7& E_8\\
+			\hline 
+			f_t& 2& 4& 6& 7& 14& 15& 15& 30\\
+			g_t& 3& 6& 9& 11& 21& 22& 23& 45\\
+			d_t& 6&12& 18& 21& 42& 44& 45& 90
+		\end{array}$$
+		and dual resolution graph.
+		
+		\
+		
+		\begin{tikzpicture}
+			
+			\draw[gray, thick] (16,-2) -- (20,-2);
+			\draw[gray, thick] (19,-2) -- (19,1);
+			\filldraw[black] (19,-1) circle (2pt) node[anchor=west]{$E_6$};
+			\filldraw[black] (19,0) circle (2pt) node[anchor=west]{$E_8$};
+			\filldraw[black] (19,1) circle (2pt) node[anchor=west]{$E_7$};
+			\filldraw[black] (16,-2) circle (2pt) node[anchor=north]{$E_1$};
+			\filldraw[black] (17,-2) circle (2pt) node[anchor=north]{$E_2$};
+			\filldraw[black] (18,-2) circle (2pt) node[anchor=north]{$E_3$};
+			\filldraw[black] (19,-2) circle (2pt) node[anchor=north]{$E_5$};
+			\filldraw[black] (20,-2) circle (2pt) node[anchor=north]{$E_4$};
+			
+			\node (W) at (19, -2) {};
+			\node (D) at (18, -1) {$d_t$};
+			\draw[->]  (W) edge (D);
+			\node (Y) at (19, 0) {};
+			\node (E) at (18, 0) {$d_t$};
+			\draw[->]  (Y) edge (E);
+			
+			
+			\node (T) at (19, -1) {};
+			\node (U) at (19, 1) {};
+			\node (FT) at (18, -0.5) {$f_1$};
+			\node (GT1) at (18, 1) {$g_1$};
+			\draw[->]  (T) edge (FT);
+			\draw[->]  (U) edge (GT1);
+			
+			\node (S) at (20, -2) {};
+			\node (GT2) at (21, -2) {$g_1$};
+			\draw[->]  (S) edge (GT2);
+			
+			
+			
+			
+		\end{tikzpicture}
+		
+		
+		
+		
+		
+		
+		
+		\
+		
+	\end{example}
+	
+	\bibliographystyle{amsplain}
+	\bibliography{biblio}
+	
+	
+	
+\end{document}
\ No newline at end of file
diff --git a/deformaciones3Edwin.ipynb b/deformaciones3Edwin.ipynb
new file mode 100644
index 0000000..f526b43
--- /dev/null
+++ b/deformaciones3Edwin.ipynb
@@ -0,0 +1,517 @@
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "f6400f90-c61b-45a3-8b92-dc7eba58c6a9",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "0"
+      ]
+     },
+     "execution_count": 1,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "load('ZetaFunctionsNewtonND.py')\n",
+    "load('zetas_triples_general.py')\n",
+    "R3.<x, y, z> = PolynomialRing(QQ, order='negdeglex')\n",
+    "R2.<u, v> = PolynomialRing(QQ, order='negdeglex')\n",
+    "f = u^4 * v^4 + u^9 + v^9\n",
+    "g = u^6 * v^6 + u^13 + v^13\n",
+    "F = z^3 - 3 * f(u=x, v=y) * z + 2 * g(u=x, v=y)\n",
+    "F.discriminant(z)(x=u, y=v) / 108 - f^3 + g^2"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "e5efc244-1749-476d-ad26-ed7cb0736fdc",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[(-1) * v^19 * (2*u^6 + v^7),\n",
+       " (-1) * u^19 * (2*v^6 + u^7),\n",
+       " (-1) * (u + v) * v^6 * u^6 * (2*u^12 - 2*u^11*v - u^10*v^2 + u^9*v^3 - u^8*v^4 + u^7*v^5 - u^6*v^6 + u^5*v^7 - u^4*v^8 + u^3*v^9 - u^2*v^10 - 2*u*v^11 + 2*v^12)]"
+      ]
+     },
+     "execution_count": 2,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "h = f^3 - g^2\n",
+    "Zh = ZetaFunctions(h)\n",
+    "[p.factor() for p in Zh.get_polyfaces_dictionary(compact=True) if p.number_of_terms() > 1]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "e0adc0c2-f648-4694-ac46-15ac55e26760",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "(u + v) * (-2*u^18*v^6 + 2*u^17*v^7 + u^16*v^8 - u^15*v^9 + u^14*v^10 - u^13*v^11 + u^12*v^12 - u^11*v^13 + u^10*v^14 - u^9*v^15 + u^8*v^16 + 2*u^7*v^17 - 2*u^6*v^18 - u^25 + u^24*v - u^23*v^2 + u^22*v^3 + 2*u^21*v^4 - 2*u^20*v^5 + 2*u^19*v^6 - 2*u^18*v^7 + 2*u^17*v^8 - 2*u^16*v^9 + 2*u^15*v^10 - 2*u^14*v^11 + 2*u^13*v^12 + 2*u^12*v^13 - 2*u^11*v^14 + 2*u^10*v^15 - 2*u^9*v^16 + 2*u^8*v^17 - 2*u^7*v^18 + 2*u^6*v^19 - 2*u^5*v^20 + 2*u^4*v^21 + u^3*v^22 - u^2*v^23 + u*v^24 - v^25 + u^26 - u^25*v + u^24*v^2 - u^23*v^3 + u^22*v^4 - u^21*v^5 + u^20*v^6 - u^19*v^7 + u^18*v^8 + 2*u^17*v^9 - 2*u^16*v^10 + 2*u^15*v^11 - 2*u^14*v^12 + 2*u^13*v^13 - 2*u^12*v^14 + 2*u^11*v^15 - 2*u^10*v^16 + 2*u^9*v^17 + u^8*v^18 - u^7*v^19 + u^6*v^20 - u^5*v^21 + u^4*v^22 - u^3*v^23 + u^2*v^24 - u*v^25 + v^26)"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "h.factor()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "f90d0141-c0d4-4ac9-bd07-8bf480ad6354",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "0"
+      ]
+     },
+     "execution_count": 4,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "load('zetas_triples_general.py')\n",
+    "R3.<x, y, z> = PolynomialRing(QQ, order='negdeglex')\n",
+    "R2.<u, v> = PolynomialRing(QQ, order='negdeglex')\n",
+    "f = (v^2 - u^3)^4 + u^2 * v^7 + v^9 + u^13\n",
+    "g = (v^2 - u^3)^6 + u^2 * v^11 + v^13\n",
+    "F = z^3 - 3 * f(u=x, v=y) * z + 2 * g(u=x, v=y)\n",
+    "F.discriminant(z)(x=u, y=v) / 108 - f^3 + g^2"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "fa1280f7-c380-49f7-953e-36aaf3bdba99",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[v^23 * (u^2 + v^2),\n",
+       " u^2 * v^7 * (v^2 - u^3)^6 * (v^4 - 6*u^3*v^2 + 3*u^6),\n",
+       " (3) * u^26 * (v^7 + u^11)]"
+      ]
+     },
+     "execution_count": 5,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "h = f^3 - g^2\n",
+    "Zh = ZetaFunctions(h)\n",
+    "[p.factor() for p in Zh.get_polyfaces_dictionary(compact=True) if p.number_of_terms() > 1]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "id": "04f7177e-79bb-43df-8dcc-f9b4c8cb394d",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "u^2*v^23 + v^25 - 12*u^5*v^21 + 2*u^4*v^22 - 12*u^3*v^23 + 4*u^2*v^24 + 2*v^26 + 54*u^8*v^19 - 12*u^7*v^20 + 55*u^6*v^21 - 24*u^5*v^22 + 3*u^4*v^23 - 12*u^3*v^24 + 3*u^2*v^25 + v^27 - 128*u^11*v^17 + 18*u^10*v^18 - 128*u^9*v^19 + 36*u^8*v^20 + 18*u^6*v^22 + 180*u^14*v^15 - 9*u^13*v^16 + 180*u^12*v^17 - 24*u^11*v^18 - 12*u^9*v^20 - 156*u^17*v^13 - 21*u^16*v^14 - 150*u^15*v^15 + 6*u^14*v^16 + 6*u^13*v^17 + 3*u^12*v^18 + 82*u^20*v^11 + 84*u^19*v^12 + 58*u^18*v^13 + 3*u^17*v^14 - 24*u^16*v^15 + 6*u^15*v^16 + 3*u^13*v^18 - 24*u^23*v^9 - 168*u^22*v^10 + 12*u^21*v^11 + 36*u^19*v^13 + 3*u^26*v^7 + 210*u^25*v^8 - 21*u^24*v^9 - 24*u^22*v^11 - 168*u^28*v^6 + 6*u^27*v^7 + 3*u^26*v^8 + 6*u^25*v^9 + 84*u^31*v^4 - 12*u^29*v^6 + 3*u^28*v^7 + 3*u^26*v^9 - 24*u^34*v^2 + 18*u^32*v^4 + 3*u^37 - 12*u^35*v^2 + 3*u^38 + u^39"
+      ]
+     },
+     "execution_count": 6,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "h.factor()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "ce0503e3-da42-46ac-bcbf-224978d2f563",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[(-1) * (u + 128*v^6)]"
+      ]
+     },
+     "execution_count": 7,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "h1 = R2(h(u=u^2, v=u^3 * (1 + v)) / u^73)\n",
+    "Zh1 = ZetaFunctions(h1)\n",
+    "[p.factor() for p in Zh1.get_polyfaces_dictionary(compact=True) if p.number_of_terms() > 1]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "id": "cf6a76e4-21df-437f-9138-9db65a93d445",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<html>\\(\\displaystyle \\left(\\begin{array}{rrrrrrrrrrrr}\n",
+       "-3 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
+       "0 & -3 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
+       "1 & 0 & -10 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\n",
+       "0 & 1 & 0 & -2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
+       "0 & 0 & 0 & 1 & -2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
+       "0 & 0 & 1 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
+       "0 & 0 & 0 & 0 & 0 & 0 & -2 & 1 & 0 & 0 & 0 & 0 \\\\\n",
+       "0 & 0 & 0 & 0 & 0 & 0 & 1 & -2 & 1 & 0 & 0 & 0 \\\\\n",
+       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -2 & 1 & 0 & 0 \\\\\n",
+       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -2 & 1 & 0 \\\\\n",
+       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -2 & 1 \\\\\n",
+       "0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1\n",
+       "\\end{array}\\right)\\)</html>"
+      ],
+      "text/latex": [
+       "$\\displaystyle \\left(\\begin{array}{rrrrrrrrrrrr}\n",
+       "-3 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
+       "0 & -3 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
+       "1 & 0 & -10 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 \\\\\n",
+       "0 & 1 & 0 & -2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
+       "0 & 0 & 0 & 1 & -2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
+       "0 & 0 & 1 & 0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n",
+       "0 & 0 & 0 & 0 & 0 & 0 & -2 & 1 & 0 & 0 & 0 & 0 \\\\\n",
+       "0 & 0 & 0 & 0 & 0 & 0 & 1 & -2 & 1 & 0 & 0 & 0 \\\\\n",
+       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -2 & 1 & 0 & 0 \\\\\n",
+       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -2 & 1 & 0 \\\\\n",
+       "0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -2 & 1 \\\\\n",
+       "0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1\n",
+       "\\end{array}\\right)$"
+      ],
+      "text/plain": [
+       "[ -3   0   1   0   0   0   0   0   0   0   0   0]\n",
+       "[  0  -3   0   1   0   0   0   0   0   0   0   0]\n",
+       "[  1   0 -10   0   0   1   0   0   0   0   0   1]\n",
+       "[  0   1   0  -2   1   0   0   0   0   0   0   0]\n",
+       "[  0   0   0   1  -2   1   0   0   0   0   0   0]\n",
+       "[  0   0   1   0   1  -1   0   0   0   0   0   0]\n",
+       "[  0   0   0   0   0   0  -2   1   0   0   0   0]\n",
+       "[  0   0   0   0   0   0   1  -2   1   0   0   0]\n",
+       "[  0   0   0   0   0   0   0   1  -2   1   0   0]\n",
+       "[  0   0   0   0   0   0   0   0   1  -2   1   0]\n",
+       "[  0   0   0   0   0   0   0   0   0   1  -2   1]\n",
+       "[  0   0   1   0   0   0   0   0   0   0   1  -1]"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "cf = [-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -10, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, -1]\n",
+    "A = Matrix(QQ, 12, cf)\n",
+    "show(A)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "id": "8d16161d-985b-4f65-a6c9-d869bc41951c",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "((0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0),\n",
+       " (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1),\n",
+       " (2, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 1))"
+      ]
+     },
+     "execution_count": 11,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "Vf, Vg, Vd = [vector(QQ, 12) for i in (1, 2, 3)]\n",
+    "Vf[9] = Vg[11] = Vd[5] = Vd[11] = 1\n",
+    "Vd[0] = Vd[2] = 2\n",
+    "Vf, Vg, Vd"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 12,
+   "id": "4551d2ce-12ce-4c6e-9c04-7bdc6bf2b8bb",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "n, Nf, Ng, Nd, Nhs, Nks, euler, nus, pols = trinomiales_general(A, Vf, Vg, (), (), Vd, total=False)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 14,
+   "id": "a2b33242-627c-44a7-a7c3-13a0ff635055",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "0 (z^3 + x^4*z^2 + x^13, (2, 1, 1), -1)\n",
+      "1 (z^3 + x^6*z^2 + x^19, (3, 1, 1), 1)\n",
+      "2 (z^3 + x^12*z^2 + x^37, (5, 1, 1), -3)\n",
+      "3 (z^3 + x^18*z^2 + x^57, (8, 1, 1), 0)\n",
+      "4 (z^3 + x^30*z^2 + x^95, (13, 1, 1), 0)\n",
+      "5 (z^3 + x^42*z^2 + x^133, (18, 1, 1), -1)\n",
+      "6 (z^3 + x^25*z + x^37, (6, 1, 1), 1)\n",
+      "7 (z^3 + x^50*z + x^74, (11, 1, 1), 0)\n",
+      "8 (z^3 + x^75*z + x^111, (16, 1, 1), 0)\n",
+      "9 (z^3 + x^100*z + x^148, (21, 1, 1), 0)\n",
+      "10 (z^3 + x^124*z + x^185, (26, 1, 1), 0)\n",
+      "11 (z^3 + x^148*z + x^222, (31, 1, 1), -1)\n",
+      "(0, 2) (z^3 + x^4*y^12*z^2 + x^13*y^37, (2, 5, 1), 1)\n",
+      "(1, 3) (z^3 + x^6*y^18*z^2 + x^19*y^57, (3, 8, 1), 1)\n",
+      "(2, 5) (z^3 + x^12*y^42*z^2 + x^37*y^133, (5, 18, 1), 1)\n",
+      "(2, 11) (z^3 + x^12*y^74*z^2 + x^37*y^222, (5, 31, 1), 1)\n",
+      "(3, 4) (z^3 + x^18*y^30*z^2 + x^57*y^95, (8, 13, 1), 1)\n",
+      "(4, 5) (z^3 + x^30*y^42*z^2 + x^95*y^133, (13, 18, 1), 1)\n",
+      "(6, 7) (z^3 + x^25*y^50*z + x^37*y^74, (6, 11, 1), 1)\n",
+      "(7, 8) (z^3 + x^50*y^75*z + x^74*y^111, (11, 16, 1), 1)\n",
+      "(8, 9) (z^3 + x^75*y^100*z + x^111*y^148, (16, 21, 1), 1)\n",
+      "(9, 10) (z^3 + x^100*y^124*z + x^148*y^185, (21, 26, 1), 1)\n",
+      "(10, 11) (z^3 + x^124*y^148*z + x^185*y^222, (26, 31, 1), 1)\n",
+      "(0, 'd') (z^3 + x^4*z^2 + x^13*y, (2, 1, 1), 2)\n",
+      "(2, 'd') (z^3 + x^12*z^2 + x^37*y, (5, 1, 1), 2)\n",
+      "(5, 'd') (z^3 + x^42*z^2 + x^133*y, (18, 1, 1), 1)\n",
+      "(11, 'd') (z^3 + x^74*z^2 + x^222*y, (31, 1, 1), 1)\n"
+     ]
+    }
+   ],
+   "source": [
+    "for a in pols:\n",
+    "    print(a, pols[a])"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 101,
+   "id": "05cb4f7a-a5d2-4533-93d2-d54c6e70268f",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "zetatop, zetamon, dic = zetas(n, pols, info=True)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 102,
+   "id": "51eca906-a9b3-467f-abb7-c3d7cc307f09",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "(216829/1151514) * (s + 35/74)^-2 * (s + 17/36)^-1 * (s + 127/266)^-1 * (s + 1)^-1 * (s^4 + 188530063/48136038*s^3 + 407335205/96272076*s^2 + 86507587/48136038*s + 34382075/128362768)"
+      ]
+     },
+     "execution_count": 102,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "zetatop.factor()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 103,
+   "id": "33a6a453-3089-40cf-81c1-0cd60c2f421b",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "{1: 1, 2: 4, 14: 1, 26: 1, 36: 1, 37: 1, 74: 4, 222: 1, 266: 1}"
+      ]
+     },
+     "execution_count": 103,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "dic"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c10b8f88-2472-4fb7-bbb0-00db0d1d0ddd",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "5a0467f7-1e57-4039-a968-1a8307e48864",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "bc0a2841-a625-4381-958e-a2b1ee373815",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "9be2fc1c-a52f-4bd8-bd3f-be31e5d26b28",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "id": "2fa88cbc-151d-4a5c-9c25-2bc9e73d13e5",
+   "metadata": {},
+   "outputs": [
+    {
+     "ename": "FeatureNotPresentError",
+     "evalue": "phitigra is not available.\nImporting GraphEditor failed: No module named 'phitigra'\nNo equivalent system packages for debian are known to Sage.\nTo install phitigra using the Sage package manager, you can try to run:\n  !sage -i phitigra\nNo equivalent system packages for pip are known to Sage.",
+     "output_type": "error",
+     "traceback": [
+      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
+      "\u001b[0;31mModuleNotFoundError\u001b[0m                       Traceback (most recent call last)",
+      "File \u001b[0;32m~/sage/src/sage/misc/lazy_import.pyx:262\u001b[0m, in \u001b[0;36msage.misc.lazy_import.LazyImport._get_object\u001b[0;34m()\u001b[0m\n\u001b[1;32m    261\u001b[0m try:\n\u001b[0;32m--> 262\u001b[0m     self._object = getattr(__import__(self._module, {}, {}, [self._name]), self._name)\n\u001b[1;32m    263\u001b[0m except ImportError as e:\n",
+      "\u001b[0;31mModuleNotFoundError\u001b[0m: No module named 'phitigra'",
+      "\nDuring handling of the above exception, another exception occurred:\n",
+      "\u001b[0;31mFeatureNotPresentError\u001b[0m                    Traceback (most recent call last)",
+      "Cell \u001b[0;32mIn[9], line 2\u001b[0m\n\u001b[1;32m      1\u001b[0m G \u001b[38;5;241m=\u001b[39m Graph(Integer(\u001b[38;5;241m12\u001b[39m))\n\u001b[0;32m----> 2\u001b[0m a \u001b[38;5;241m=\u001b[39m \u001b[43mgraph_editor\u001b[49m\u001b[43m(\u001b[49m\u001b[43mG\u001b[49m\u001b[43m)\u001b[49m\n",
+      "File \u001b[0;32m~/sage/src/sage/misc/lazy_import.pyx:409\u001b[0m, in \u001b[0;36msage.misc.lazy_import.LazyImport.__call__\u001b[0;34m()\u001b[0m\n\u001b[1;32m    407\u001b[0m         True\n\u001b[1;32m    408\u001b[0m     \"\"\"\n\u001b[0;32m--> 409\u001b[0m     return self.get_object()(*args, **kwds)\n\u001b[1;32m    410\u001b[0m \n\u001b[1;32m    411\u001b[0m def __repr__(self):\n",
+      "File \u001b[0;32m~/sage/src/sage/graphs/graph_editor.py:81\u001b[0m, in \u001b[0;36mgraph_editor\u001b[0;34m(graph, **display_options)\u001b[0m\n\u001b[1;32m     78\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m graph \u001b[38;5;129;01mis\u001b[39;00m \u001b[38;5;28;01mNone\u001b[39;00m:\n\u001b[1;32m     79\u001b[0m     graph \u001b[38;5;241m=\u001b[39m Graph(\u001b[38;5;241m0\u001b[39m)\n\u001b[0;32m---> 81\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43mGraphEditor\u001b[49m\u001b[43m(\u001b[49m\u001b[43mgraph\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[38;5;241;43m*\u001b[39;49m\u001b[43mdisplay_options\u001b[49m\u001b[43m)\u001b[49m\n",
+      "File \u001b[0;32m~/sage/src/sage/misc/lazy_import.pyx:409\u001b[0m, in \u001b[0;36msage.misc.lazy_import.LazyImport.__call__\u001b[0;34m()\u001b[0m\n\u001b[1;32m    407\u001b[0m         True\n\u001b[1;32m    408\u001b[0m     \"\"\"\n\u001b[0;32m--> 409\u001b[0m     return self.get_object()(*args, **kwds)\n\u001b[1;32m    410\u001b[0m \n\u001b[1;32m    411\u001b[0m def __repr__(self):\n",
+      "File \u001b[0;32m~/sage/src/sage/misc/lazy_import.pyx:225\u001b[0m, in \u001b[0;36msage.misc.lazy_import.LazyImport.get_object\u001b[0;34m()\u001b[0m\n\u001b[1;32m    223\u001b[0m     if likely(self._object is not None):\n\u001b[1;32m    224\u001b[0m         return self._object\n\u001b[0;32m--> 225\u001b[0m     return self._get_object()\n\u001b[1;32m    226\u001b[0m \n\u001b[1;32m    227\u001b[0m cpdef _get_object(self) noexcept:\n",
+      "File \u001b[0;32m~/sage/src/sage/misc/lazy_import.pyx:267\u001b[0m, in \u001b[0;36msage.misc.lazy_import.LazyImport._get_object\u001b[0;34m()\u001b[0m\n\u001b[1;32m    265\u001b[0m     # Avoid warnings from static type checkers by explicitly importing FeatureNotPresentError.\n\u001b[1;32m    266\u001b[0m     from sage.features import FeatureNotPresentError\n\u001b[0;32m--> 267\u001b[0m     raise FeatureNotPresentError(feature, reason=f'Importing {self._name} failed: {e}')\n\u001b[1;32m    268\u001b[0m raise\n\u001b[1;32m    269\u001b[0m \n",
+      "\u001b[0;31mFeatureNotPresentError\u001b[0m: phitigra is not available.\nImporting GraphEditor failed: No module named 'phitigra'\nNo equivalent system packages for debian are known to Sage.\nTo install phitigra using the Sage package manager, you can try to run:\n  !sage -i phitigra\nNo equivalent system packages for pip are known to Sage."
+     ]
+    }
+   ],
+   "source": [
+    "G = Graph(12)\n",
+    "a = graph_editor(G)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 54,
+   "id": "54c20ec5-3e80-4d4d-9586-2af443c62f95",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "application/vnd.jupyter.widget-view+json": {
+       "model_id": "62e83ac19ea047aeb488ebd6c6a0bad9",
+       "version_major": 2,
+       "version_minor": 0
+      },
+      "text/plain": [
+       "HBox(children=(VBox(children=(MultiCanvas(height=400, image_data=b'\\x89PNG\\r\\n\\x1a\\n\\x00\\x00\\x00\\rIHDR\\x00\\x00\\x02X\\x00\\x00\\x01\\x90\\x08\\x06\\x00\\x00\\x00r5\\x1e\\x98\\x00\\x00\\x00\\x01sRGB\\x00\\xae\\xce\\x1c\\xe9\\x00\\x00 \\x00IDATx^\\xed\\xdd\\t|T\\xd5\\xf9\\xff\\xf1\\xefL\\x12\\x92I\\xc2\\x16\\x08\\xfb\\xbe\\xc8\\xbe(n n\\x88\\x8a\\x88\\n\\x8a\\xe2\\xbeV\\xad\\xd5V\\xdbZ\\xabmm\\xb5j\\x7fU\\xab\\xd5V\\xebn]qA\\xc5\\x15A\\x14EQT@Y$l\\xb2\\xcaN\\x08[\\x92\\xc9>\\xff\\x9e\\x13&\\x06%\\xc9\\xdc\\x99\\x9b\\xc9\\x1d\\xff\\x9f\\xfbz\\xe57\\xf6\\xc7\\xe4\\xccy\\xde\\xcf\\xb93O\\xe6\\x9e{\\x8eO\\x1c\\x08 \\x80\\x00\\x02\\x08 \\x80\\x00\\x02\\xae\\n\\xf8\\\\m\\x8d\\xc6\\x10@\\x00\\x01\\x04\\x10@\\x00\\x01\\x04D\\x81\\xc5 @\\x00\\x01\\x04\\x10@\\x00\\x01\\x04\\\\\\x16\\xa0\\xc0r\\x19\\x94\\xe6\\x10@\\x00\\x01\\x04\\x10@\\x00\\x01\\n,\\xc6\\x00\\x02\\x08 \\x80\\x00\\x02\\x08 \\xe0\\xb2\\x00\\x05\\x96\\xcb\\xa04\\x87\\x00\\x02\\x08 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\\xb1\\x05(\\xb0\\x12;\\x7f\\xf4\\x1e\\x01\\x04\\x10@\\x00\\x01\\x04<(@\\x81\\xe5\\xc1\\xa4\\xd0%\\x04\\x10@\\x00\\x01\\x04\\x10Hl\\x01\\n\\xac\\xc4\\xce\\x1f\\xbdG\\x00\\x01\\x04\\x10@\\x00\\x01\\x0f\\nP`y0)t\\t\\x01\\x04\\x10@\\x00\\x01\\x04\\x12[\\x80\\x02+\\xb1\\xf3G\\xef\\x11@\\x00\\x01\\x04\\x10@\\xc0\\x83\\x02\\x14X\\x1eL\\n]B\\x00\\x01\\x04\\x10@\\x00\\x81\\xc4\\x16\\xa0\\xc0J\\xec\\xfc\\xd1{\\x04\\x10@\\x00\\x01\\x04\\x10\\xf0\\xa0\\x00\\x05\\x96\\x07\\x93B\\x97\\x10@\\x00\\x01\\x04\\x10@ \\xb1\\x05(\\xb0\\x12;\\x7f\\xf4\\x1e\\x01\\x04\\x10@\\x00\\x01\\x04<(\\xf0\\xff\\x00+_\\xd4\\x08;\\xc2P(\\x00\\x00\\x00\\x00IEND\\xaeB`\\x82', layout=Layout(border_bottom='3px solid lightgrey', border_left='3px solid lightgrey', border_right='3px solid lightgrey', border_top='3px solid lightgrey', height='406px', overflow='visible', width='606px'), sync_image_data=True, width=600), HBox(children=(Label(value='Graph on 12 vertices and 0 edges.', layout=Layout(width='100%')), Label(value='Graph Editor', layout=Layout(width='100%')))), Output())), VBox(children=(ToggleButtons(icons=('', '', '', '', ''), layout=Layout(margin='0px 2px 0px auto', width='150px'), options=('select / move', 'add vertex or edge', 'delete vertex or edge', 'add walk', 'add clique', 'add star'), tooltips=('Move vertices or the canvas', 'Add vertices or edges', 'Delete vertices or edges', 'Add a walk through new or existing vertices', 'Add a clique through new or existing vertices', 'Add a star through new or existing vertices'), value='select / move'), Dropdown(layout=Layout(margin='1px 2px 1px auto', width='150px'), options=(('- change layout -', '- change layout -'), ('random', 'random'), ('spring', 'spring'), ('circular', 'circular'), ('planar', 'planar'), ('forest (root up)', 'forest (root up)'), ('forest (root down)', 'forest (root down)'), ('directed acyclic', 'acyclic')), value='- change layout -'), HBox(children=(Button(icon='search-plus', layout=Layout(height='36px', margin='0px 1px 0px 0px', width='36px'), style=ButtonStyle(), tooltip='Zoom in'), Button(icon='compress', layout=Layout(height='36px', margin='0px 1px 0px 1px', width='36px'), style=ButtonStyle(), tooltip='Zoom to fit'), Button(icon='search-minus', layout=Layout(height='36px', margin='0px 1px 0px 1px', width='36px'), style=ButtonStyle(), tooltip='Zoom out'), Button(icon='trash', layout=Layout(height='36px', margin='0px 0px 0px 1px', width='36px'), style=ButtonStyle(), tooltip='Clear the drawing and delete the graph')), layout=Layout(margin='1px 2px 1px auto')), HBox(children=(ColorPicker(value='#437FC0', layout=Layout(height='36px', margin='0px 1px 0px 0px', width='112px')), Button(icon='paint-brush', layout=Layout(height='36px', margin='0px 0px 0px 1px', width='36px'), style=ButtonStyle(), tooltip='Apply color to the selected elements')), layout=Layout(margin='1px 2px 1px auto')), HBox(children=(Label(value='⌀', layout=Layout(margin='auto 2px auto 0px')), BoundedIntText(value=20, layout=Layout(margin='auto 1px auto 0px', width='90px'), min=1), Button(icon='expand', layout=Layout(height='36px', margin='0px 0px 0px 1px', width='36px'), style=ButtonStyle(), tooltip='Change the radius of the selected vertices')), layout=Layout(margin='1px 2px 1px auto')), ToggleButton(value=True, description='Show vertex labels', layout=Layout(margin='1px 2px 1px auto', width='150px'), tooltip='Should the vertex label be drawn?'), ToggleButton(value=True, description='Show edge labels', layout=Layout(margin='1px 2px 1px auto', width='150px'), tooltip='Should the edge label be drawn?')), layout=Layout(min_width='160px', width='160px'))), layout=Layout(height='auto', overflow='auto hidden', width='100%'))"
+      ]
+     },
+     "execution_count": 54,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "a.show()"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 91,
+   "id": "eed1947a-35d3-4741-a7ea-f8550a2abd30",
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "1\n",
+      "[-3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, -10, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, -1]\n"
+     ]
+    }
+   ],
+   "source": [
+    "A = G.adjacency_matrix()\n",
+    "A[0, 0] = A[1, 1] = -3\n",
+    "A[2, 2] = -10\n",
+    "A[5, 5] = A[11, 11]= -1\n",
+    "for i in (3, 4) + tuple(6, .., 10):\n",
+    "    A[i, i] = -2\n",
+    "print(A.det())\n",
+    "print(A.list())"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "b174f66f-0e7e-4b03-9e22-3d10858d6cef",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "SageMath 10.3.beta4",
+   "language": "sage",
+   "name": "sagemath"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.11.6"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}