olvido

DeformacionGenerica_0.ipynb

@@ -5,7 +5,23 @@
    "execution_count": 1,
    "id": "72d5a30c-1a59-4a55-9680-fdfa7979625b",
    "metadata": {},
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<html>\\(\\displaystyle z^{3} - 3 y^{2} z + 3 x^{2} z + 2 x^{4}\\)</html>"
+      ],
+      "text/latex": [
+       "$\\displaystyle z^{3} - 3 y^{2} z + 3 x^{2} z + 2 x^{4}$"
+      ],
+      "text/plain": [
+       "z^3 - 3*y^2*z + 3*x^2*z + 2*x^4"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
    "source": [
     "# load('zetas_triples_general.py')\n",
     "R.<x, y, z> = PolynomialRing(QQ, order='neglex')\n",
@@ -13,7 +29,8 @@
     "f = v^2 - u^2\n",
     "g = u^4\n",
     "F = z^3 - 3 * f(u=x, v=y) * z + 2 * g(u=x, v=y)\n",
-    "h = F.discriminant(z)(x=u, y=v)"
+    "h = F.discriminant(z)(x=u, y=v)\n",
+    "show(F)"
    ]
   },
   {
@@ -72,6 +89,21 @@
    "id": "27e92ddc-34d6-4290-880f-9d3561a8ec4e",
    "metadata": {},
    "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<html>\\(\\displaystyle z^{3} - 3 y^{2} z + t x z^{2} + 3 x^{2} z + 2 x^{4}\\)</html>"
+      ],
+      "text/latex": [
+       "$\\displaystyle z^{3} - 3 y^{2} z + t x z^{2} + 3 x^{2} z + 2 x^{4}$"
+      ],
+      "text/plain": [
+       "z^3 - 3*y^2*z + t*x*z^2 + 3*x^2*z + 2*x^4"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
     {
      "data": {
       "text/plain": [
@@ -85,12 +117,40 @@
    ],
    "source": [
     "Ft = F + t * x * z^2\n",
+    "show(Ft)\n",
     "Ft.jacobian_ideal().vector_space_dimension()"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": 5,
+   "id": "83328b3a-8da8-4adc-8b6c-02764b1420d3",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<html>\\(\\displaystyle 108 y^{6} + \\left(9 t^{2} - 324\\right) x^{2} y^{4} + \\left(-18 t^{2} + 324\\right) x^{4} y^{2} - 108 t x^{5} y^{2} + \\left(9 t^{2} - 108\\right) x^{6} + \\left(-8 t^{3} + 108 t\\right) x^{7} - 108 x^{8}\\)</html>"
+      ],
+      "text/latex": [
+       "$\\displaystyle 108 y^{6} + \\left(9 t^{2} - 324\\right) x^{2} y^{4} + \\left(-18 t^{2} + 324\\right) x^{4} y^{2} - 108 t x^{5} y^{2} + \\left(9 t^{2} - 108\\right) x^{6} + \\left(-8 t^{3} + 108 t\\right) x^{7} - 108 x^{8}$"
+      ],
+      "text/plain": [
+       "108*y^6 + (9*t^2 - 324)*x^2*y^4 + (-18*t^2 + 324)*x^4*y^2 + (-108*t)*x^5*y^2 + (9*t^2 - 108)*x^6 + (-8*t^3 + 108*t)*x^7 - 108*x^8"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "ht = Ft.discriminant(z)\n",
+    "show(ht)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
    "id": "8cc358aa-03d1-4b9c-b4ff-87b0324bd55b",
    "metadata": {},
    "outputs": [],
@@ -101,7 +161,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 8,
+   "execution_count": 7,
    "id": "d21a93b0-b0c4-4cd2-b648-f4bca8f78376",
    "metadata": {},
    "outputs": [],
@@ -112,7 +172,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 11,
+   "execution_count": 8,
    "id": "59240908-d672-4520-90e5-7358840f8c23",
    "metadata": {},
    "outputs": [
@@ -122,7 +182,7 @@
        "[(-108) * (-v + u)^3 * (v + u)^3]"
       ]
      },
-     "execution_count": 11,
+     "execution_count": 8,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -133,7 +193,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 12,
+   "execution_count": 9,
    "id": "a1a4111c-050d-4656-be38-ceac87684b47",
    "metadata": {},
    "outputs": [
@@ -143,7 +203,7 @@
        "[(9) * (v - u)^2 * (v + u)^2 * (12*v^2 + (t^2 - 12)*u^2)]"
       ]
      },
-     "execution_count": 12,
+     "execution_count": 9,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -154,7 +214,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 13,
+   "execution_count": 10,
    "id": "8ac2847d-121a-4a78-a202-7e1692998c99",
    "metadata": {},
    "outputs": [],
@@ -165,7 +225,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 15,
+   "execution_count": 11,
    "id": "e7884c6b-458b-4d7c-8c8f-376ec4b87c9a",
    "metadata": {},
    "outputs": [
@@ -175,7 +235,7 @@
        "[(108) * v^3 * (v + 2*u)^3, (108) * u^3 * (8*v^3 - u^5)]"
       ]
      },
-     "execution_count": 15,
+     "execution_count": 11,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -186,7 +246,7 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 16,
+   "execution_count": 12,
    "id": "442b2c9c-c7f1-4fb9-ac12-bf372d90a342",
    "metadata": {},
    "outputs": [
@@ -197,7 +257,7 @@
        " ((4*t^2)) * u^4 * (9*v^2 + (-2*t)*u^3)]"
       ]
      },
-     "execution_count": 16,
+     "execution_count": 12,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -260,6 +320,46 @@
     "[p.factor() for p in Ztb.get_polyfaces_dictionary(compact=True) if p.number_of_terms() > 1]"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": 16,
+   "id": "ea740cbb-5980-45e4-a6c9-695a78324c50",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "Zt1 = ZetaFunctions(ht1)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "id": "d98d3aec-353c-4dcf-9b8a-1ea5ca93512e",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[((-4*t^2)) * (-9*v^2 + (2*t)*u)]"
+      ]
+     },
+     "execution_count": 17,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "ht1 = S(ht(v=v + u)(v=u * v) / u^6)\n",
+    "[p.factor() for p in Zt1.get_polyfaces_dictionary(compact=True) if p.number_of_terms() > 1]"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "c78e9c40-ee08-4a4b-a0ca-4de12082d245",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
   {
    "cell_type": "code",
    "execution_count": 24,

Mult3.tex

@@ -401,7 +401,7 @@ The singular point is isolated if and only if the discriminant $D_zF$ is reduced
 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 $c_3=(\frac{-b}{2})^3$.}
+\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 }
 
@@ -445,7 +445,7 @@ $N^\Delta_j - 3 N^0_j = N^\Delta_j - 2 N^\infty_j$ is called the \emph{excess of
 \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.  
+$b_3= (y^3+x^4)(x^3+y^4)$. The minimal resolution $\pi$ of the product $G=b_2 b_3 D_zF$ is obtained after 17 blow-ups.  The following figure and table give the dual resolution graph of $G$ and its associated numerical data.  
 
 
 
@@ -587,6 +587,9 @@ N^g_i& {\bf12}& {\bf 13}& {\bf 26}& {\bf 39}& {\bf 52}& {\bf 65}& {\bf 78}& {\bf
 N^\Delta_i& 25& {\bf 26}& {\bf 52}& {\bf 78}& {\bf 104}& {\bf 130}& {\bf 156}& {\bf 52}&{\bf 104}\\
 \hline
 \hline
+\pi()& 1& 0& 0& 0& 0& 0& \mathbb{P}^1& 0&0\\
+\hline
+\hline
 \nu_i&2&3&5&7&9&11&13&6&11\\
 \chi()&-13&1&-1&0&0&0&-2&1&-1\\
 E_i^2& -11& -2& -4& -2& -2& -2& -1&-2&-1\\
@@ -676,7 +679,9 @@ in the case $I= \{i, j\}$.
 \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}
 
 
-$$\begin{array}{c|c|c|c}
+
+
+  $$\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\\
@@ -688,7 +693,6 @@ i&f_i&\omega_i& \chi()\\
 \end{array}$$
 
 
-
 $$\begin{array}{c|c|c}
 (i,j)&f_{i,j}&\omega_{i,j}\\
 \hline
@@ -1044,18 +1048,22 @@ Therefore, $F$ is non degenerate with respect to its Newton polyhedron if and on
  
  \begin{proposition}\label{formestrpi}
 The monodromy zeta function $\zeta_F(t)$ and the local topological  zeta function $Z(F,s)_0$ of $F$ equal
-$$(1 - t^3)^{\alpha} \prod_{I} \zeta_{f_I}(t)^{-\chi(E_I^\circ)}$$ 
+$$(1 - t^3)^{-\alpha} \prod_{I} \zeta_{f_I}(t)^{\chi(E_I^\circ)}$$ 
 $$\sum_I \chi(E^\circ_I) Z(f_I, \omega_I)_{o_I}$$
  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. 
+ \begin{proof} It follows from the formulae for $\zeta_f$ in terms of a resolution \cite{AC}, in terms of  a modification \cite[Theorem 1]{GSLM97}, and the  Stratum  principle for topological zeta functions \cite[Stratum Principle 1.2]{ABCNLM02}. Let us discuss the case of $\zeta_F$. The Milnor fiber $M_F$ intersects  the $z$-axis in a set $A$ of three points that it is invariant under the action of the Milnor monodromy, i.e., the three points are cyclically permuted. Hence, we have $$\zeta_F = \zeta_F_{|_{M_F \setminus A}} \cdot  \zeta_F_{|_A}  = \zeta_F_{|_{M_F \setminus A}} (1 - t^3) .$$
+ In order to consider compute the term $\zeta_F_{|_{M_F \setminus A}} $ we consider the modification  $\tilde{X} : =X \times \CC \rightarrow \CC^{3}$. Denote the pull back of the Milnor fibre under $\tilde{\pi}$ by $\tilde{M}$. Assume that $\pi$ is a composition of $\alpha$ point blow-ups. Denote by $B$ the intersection of $\tilde{M}$ with the $\alpha$ vertical planes thru the exceptional divisors of $\pi$. The set $B$ is invariant under the pull back of the Milnor monodromy. Since the exceptional locus of $\pi$ is a tree of $\alpha$ exceptional divisors isomorphic to $\mathbb{P}^1$, its Euler characteristic is  $\alpha + 1$. Therefore,
+$$\zeta_F= \zeta_{\pi^\ast F} = \zeta_{\pi^\ast F_{|_{\tilde{M} \setminus B}}}  \zeta_{\pi^\ast F_{|_B}} = \zeta_{\pi^\ast F_{|_{\tilde{M} \setminus B}}}  (1 - t^3)^{a+1}.$$
+ Finally, we can express  $\zeta_{\pi^\ast F} = \prod_{I} \zeta_{f_I}(t)^{\chi(E_I^\circ)}.$$
+ % 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
+\begin{example} The resolution $\pi$ of Example \ref{extriplepole} consist of $\alpha=17$ blow-ups and the associated stratification has 35 strata. According to Proposition \ref{formestrpi} and using the simmetry of the resolution graph and the fact that the monodromy zeta function coincides for the 16 points of intersection of two different exceptional divisor, 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_F(t) = \frac{\zeta^{2}_{f_{2}}   \zeta^{2}_{f_{8}}  \zeta^{16}_{f_{1,7}} \zeta^{13}_{f_{1,16}} \zeta^{2}_{f_{7,29}} \zeta^{2}_{f_{7,33}} \zeta^{2}_{f_{9,31}}  }
-{\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}$$
+{(1-t^3)^{17}  \zeta^{13}_{f_{1}}  \zeta^{2}_{f_{3}}  \zeta^{4}_{f_{7}} \zeta^{2}_{f_{9}}  }= \Phi^3_1 \Phi^{15}_2 \Phi_3 \Phi^2_6 \Phi^2_{13} \Phi^{15}_{26} \Phi^2_{78}$$
 where $\Phi_i$ is $i$-th the cyclotomic polymonial, and $Z(F,s)_0$ equals 
 $$-13Z(f_1,s)_0 + \sum_{i=2,8,10,16}Z(f_i,s)_0 -  \sum_{i=3, 9,11,17} Z(f_i,s)_0 - 2 \sum_{i=7,15}Z(f_i,s)_0 + \sum_{(i,j) \in S} Z(f_{i,j},s)_0=$$
 $$\frac{-1/26s^3 + 2749/6084s^2 + 1969/4056s + 1/8}{(s+1)(s+1/2)^3}$$
@@ -1305,8 +1313,68 @@ $$Z(F_7, \omega_7)_{o_7}=Z(F_{7,33}, \omega_{7,33})_{o_{7,33}}= Z(F_{15}, \omega
     
   \subsection{Third reduction: $\QQ$-resolution of the discriminant}
   
+  
+  
+
+
+$$\begin{array}{c|c|c|c|c}
+i&f_i&\omega_i& \chi()& \zeta_{f_i}\\
+\hline
+1& z^3 + x^4z^2 + x^{13}& (2, 1, 1)&  -13&  ( \Phi_2^2 \Phi_4 \Phi_6 \Phi_{12} \Phi_{26})^{-1} \\
+2,10& z^3 + x^{10}z + x^{13}& (3, 1, 1)&1&  \Phi_1 (\Phi_{39})^{-1}\\
+3,11& z^3 + x^{20}z + x^{26}& (5, 1, 1)& -1&  \Phi_1 ({\Phi_6 \Phi_{39} \Phi_{78})^{-1}\\
+7,15& z^3 + \color{red}{x^{52}z} + x^{78}& (13, 1, 1)& -2&( \Phi_1 \Phi_2^2 \textcolor{red}{\Phi_3} \Phi_6^2 \Phi_{13}^2 \Phi_{26}^2 \Phi_{39}^2 \Phi_{78}^2 )^{-1}\\
+8, 16&  z^3 + x^{21}z + x^{26}& (6, 1, 1)& 1&  \Phi_1 (\Phi_6 \Phi_{39} \Phi_{78})^{-1}\\
+ 9,17& z^3 + x^{42}z + x^{52}& (11, 1, 1)& -1& \Phi_1 (\Phi_6 \Phi_{12} \Phi_{39} \Phi_{78} \Phi_{156})^{-1}\\
+\end{array}$$
+
+$$\begin{array}{c|c|c|c}
+(i,j)&f_{i,j}&\omega_{i,j}\\
+\hline
+&&\\
+(1, 7), (1, 15)& z^3 + x^4y^{26}z^2 + x^{13}y^{78}& (2, 13, 1)& \Phi_1 \textcolor{red}{\Phi_3} \\
+(2,3),  (10, 11)& z^3 + x^{10}y^{20}z + x^{13}y^{26}& (3, 5, 1)& \Phi_1 \textcolor{red}{\Phi_3} \\
+(3, 9), (11, 17)& z^3 + x^{20}y^{42}z + x^{26}y^{52}& (5, 11, 1)& \Phi_1 \textcolor{red}{\Phi_3} \\
+(3, 4), (11,12)& z^3 + x^{20}y^{28}z + x^{26}y^{39}& (5, 7, 1)& \Phi_1 \textcolor{red}{\Phi_3} \\
+(8, 9), (16,17)& z^3 + x^{21}y^{42}z + x^{26}y^{52}& (6, 11, 1)& \Phi_1 \textcolor{red}{\Phi_3} \\
+(4, 5), (12,13)& z^3 + x^{28}y^{36}z + x^{39}y^{52}& (7, 9, 1)& \Phi_1 \textcolor{red}{\Phi_3} \\
+(5, 6), (13,14)& z^3 + x^{36}y^{44}z + x^{52}y^{65}& (9, 11, 1)& \Phi_1 \textcolor{red}{\Phi_3} \\
+(6, 7), (14,15)& z^3 + x^{44}y^{52}z + x^{65}y^{78}& (11, 13, 1)& \Phi_1 \textcolor{red}{\Phi_3}\\
+(1, i), i=16, \dots, 28&  z^3 + x^4z^2 + x^{13}y& (2, 1, 1)& (\Phi_2 \Phi_4 \Phi_6 \Phi_{12})^{-1} \\
+(9, 31), (17, 32)& z^3 + x^{42}yz + x^{52}& (11, 1, 1)&\Phi_1  (\Phi_6 \Phi_{12}  \Phi_{39}  \Phi_{78}  \Phi_{156})^{-1} \\
+(7, 33), (15, 34)& z^3 + x^{52}z + x^{78}y& (13, 1, 1)&  (\Phi_1 \Phi_2^2 \textcolor{red}{\Phi_3}  \Phi_6^2 \Phi_{13}^2  \Phi_{26}^2   \Phi_{39}^2 \Phi_{78}^2 )^{-1} \\
+(7, 29) (15, 30)& z^3 + x^{26}z^2 + x^{78}y& (13, 1, 1)& (\Phi_2  \Phi_6 \Phi_{13}  \Phi_{26}   \Phi_{39} \Phi_{78})^{-1} \\
+%(12, 'f')& z^3 + x^{42}yz + x^{52}& (11, 1, 1)\\
+%(16, 'g')& z^3 + x^{52}z + x^{78}y& (13, 1, 1)\\
+%(16, 'd')& z^3 + x^{26}z^2 + x^{78}y&  (13, 1, 1)
+\end{array}$$
+  
+  
+$$\begin{align}
+\Phi_1& =& 17 - 13  + 8 - 16 - 2 &=& 18\\
+\Phi_2 &=& 26 + 8 -13 -2 -4  &=& 15\\
+\Phi_3 &=& -17  + 4  + 16  - 2 &= &1\\
+\Phi_4 &=& 13 -13 &=& 0\\
+\Phi_6 &=& 13 +2 + 8 -2  + 2 -13 -2-2-4 &=& 2\\
+\Phi_{12} &=& 13 + 2 -13 -2 &=& 0\\
+\Phi_{13} &= &8  -2  - 4& =& 2\\
+\Phi_{26} &= &13 + 8 -2 -4  & =& 15\\
+\Phi_{39} &= &-2 +  2  - 8  -2 + 2   +  2      + 2 + 4& =& 0 \\
+\Phi_{78} &= &2 + 8 -2 + 2  - 2 - 2 - 4& =& 2\\
+\Phi_{156} &= & 2 - 2& = &0 
+\end{align}$$
+
+
+$$\Phi_1^{18}\Phi_2^{15}\Phi_3\Phi_6^2\Phi_{13}^2\Phi_{26}^{15}\Phi_{78}^2$$
+
+
+
  \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. 
  
  
@@ -1533,6 +1601,10 @@ and dual resolution graph.
  
  \end{example}
  
+ 
+ 
+ 
+ 
 \bibliographystyle{amsplain}
 \bibliography{biblio}