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@ -310,13 +310,13 @@ Consider a the germ of a multiplicity 3 singular point of hypersurface. An equat
\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 ase of a suspension that we have studied elsewhere. See \cite{}.
\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. 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 $f \in \CC[x,y]$.
\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
@ -402,6 +402,8 @@ The monic Weierstrass polynomial $z^3 -3b_2z + 2b_3$ is non degenerate unless $M
\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
$$E_I^\circ = \bigcap_{i \in I} E_j \setminus \bigcup_{j \not \in I} E_j .$$
@ -421,12 +423,17 @@ There are four possibilities for the behaviour of the numbers $N^0_j, N^\infty_j
\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}
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.
\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} For a divisor of the last type, the difference
\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}
@ -435,7 +442,7 @@ $N^\Delta_j - 3 N^0_j = N^\Delta_j - 2 N^\infty_j$ 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 15 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 30 blow-ups. The following figure and table give the dual resolution graph of $G$ and its associated numerical data.
@ -453,7 +460,7 @@ $b_3= (y^3+x^4)(x^3+y^4)$. The minimal resolution $\pi$ of the product $G=b_2 b_
\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 ${{F^\Delta_i}}_{{}_{i=1}}^{{}{13}}$}};
\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);
@ -485,17 +492,17 @@ $b_3= (y^3+x^4)(x^3+y^4)$. The minimal resolution $\pi$ of the product $G=b_2 b_
\coordinate (F1) at (1.2,0.75);
\node[above] at (F1) {{\small $F^{\infty}_1$}};
\node[above] at (F1) {{\small $F^{\infty}_{33}$}};
\coordinate (F2) at (1.95,0.75);
\node[above] at (F2) {{\small $F^\Delta_{14}$}};
\node[above] at (F2) {{\small $F^\Delta_{29}$}};
\coordinate (F3) at (6,0.75);
\node[above] at (F3) {{\small $F^{0}_1$}};
\node[above] at (F3) {{\small $F^{0}_{31}$}};
\coordinate (F4) at (-1.2,0.75);
\node[above] at (F4) {{\small $F^{\infty}_2$}};
\node[above] at (F4) {{\small $F^{\infty}_{34}$}};
\coordinate (F5) at (-1.95,0.75);
\node[above] at (F5) {{\small $F^\Delta_{15}$}};
\node[above] at (F5) {{\small $F^\Delta_{30}$}};
\coordinate (F6) at (-6,0.75);
\node[above] at (F6) {{\small $F^{0}_2$}};
\node[above] at (F6) {{\small $F^{0}_{32}$}};
@ -578,42 +585,45 @@ 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()&-13&1&-1&2&2&2&-4&1&-1\\
\chi()&-11&1&-1&0&0&0&-2&1&-1\\
E_i^2& -7& -2& -2& -2& -2& -2& -1&-2&-1\\
\hline
\end{array}$$
The divisor $E_1$ is mapped to the value 1, the divisors $E_7$ and $E_{13}$ are dicriticals, and the rest of exceptionals divisors are poles. The 13 arrows $F^\Delta_i$ with $i=1, \dots, 13$ attached to the divisor $E_1$ correspond to irreducible components of the strict transforms of $D_zF$, the arrows attachef to the divisor $E_7$ (resp. $E_{13}$) correspond to an irreducible component strict transform of $b_3$ and another irreducible component of $D_zF$.
The divisor $E_1$ is mapped to the value 1, the divisors $E_7$ and $E_{15}$ are dicriticals, and the rest of exceptionals divisors are zeros. The 13 arrows $F^\Delta_i$ with $i=16, \dots, 29$ attached to the divisor $E_1$ correspond to irreducible components of the strict transforms of $D_zF$, the arrows attached to the divisor $E_7$ (resp. $E_{13}$) correspond to an irreducible component strict transform of $b_3$ and another irreducible component of $D_zF$. Finally the arrow attched to $E_9$ (resp. to $E_{17}$) correspond to the strict transform of $b_2$.
%$$\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_{1,2}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}_1 + 3 F^{\infty}_2, \quad \pi(E_7) = \pi(E_{15}) = \mathbb{P}^1, \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$$
The stratification associated with the resolution $\pi$ consist of $35$ strata $E^\circ_{i,j}$ of dimension 0 and 8 strata $E_i^\circ$ of dimension 1 and $\chi(E_i^\circ) \ne 0$.
The former correspond to the 16 pairs $(i,j)$ listed belllow
$$\begin{array}{cccccccc}
(1,7)& (2,3)& (3,4)& (3,9)& (4,5)& (5,6)& (6,7)& (8, 9) \\
(1,15)& (10,11)& (11,12)& (11,17)& (12, 13)& (13,14)& (14,15)& (16,17)
\end{array}$$
the 13 intersections $E_1 \cap F_i^\Delta$ for $i=1, \dots, 13$ and the 6 intersections
$$ E_7 \cap F_1^\infty, \quad E_7 \cap F_{14}^\Delta, \quad E_9 \cap F_1^0, \quad E_{15} \cap F_2^\infty, \quad E_{15} \cap F_{14}^\Delta, \quad E_{17} \cap F_2^0 $$
The stratification associated with the resolution $\pi$ consist of $35$ strata $E^\circ_{i,j}$ of dimension 0 and 11 strata $E_i^\circ$ of dimension 1 and $\chi(E_i^\circ) \ne 0$.
The former correspond to the elements $(i,j)$ of the set
$$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 10, \quad 11, \quad 15.$$
$$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}
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 $\pi^\ast F$ can be expressed in terms of the pullbacks of $b_2$ and $b_3$ as
\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$,
for the pullback of $F$ under the birational map $\pi \times id$.
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^f_i$ and $N^g_i$ with $i \in I$ as
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)
@ -632,11 +642,11 @@ If the strata $E^\circ_{i,j} \ne \emptyset$ and $\mathcal{M}(E_i)=1$ then either
\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$$
$$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)$$
if both $\mathcal{M}(E_i) = \mathcal{M}(E_j)=1$
$$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)$$
if both $\mathcal{M}(E_i) = \mathcal{M}(E_j)=1$
or
$$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)$$
if $\mathcal{M}(E_i)=1$ and $E_j$ is dicritical
@ -652,7 +662,7 @@ in the case $I= \{i\}$
and
$$\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_ij^\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 + 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}$$
@ -988,9 +998,9 @@ Therefore, $F$ is non degenerate with respect to its Newton polyhedron if and on
\section{Contibutions to the topological zeta function and the A´Campo monodromy zeta function of the trinomial forms}
\begin{proposition}
The monodromy zeta function and the local topological zeta function $Z(F,s)_0$ of $F$ equal
$$(1 - t^2)^{\alpha} \prod_{I} \zeta_{f_I}(t)^{-\chi(E_I^\circ)}$$
\begin{proposition}\label{formestrpi}
The monodromy zeta function $\zeta_F(t)$ and the local topological zeta function $Z(F,s)_0$ of $F$ equal
$$(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}$$
where $I$ varies over the statification associated to the resolution $\pi$ of $b_2b_3D_zF$, the exponent $\alpha$ is the number of points of blow-ups needed to obtain $\pi$, and $o_I$ is a point in $E^\circ_I$.
@ -998,9 +1008,12 @@ $$\sum_I \chi(E^\circ_I) Z(f_I, \omega_I)_{o_I}$$
\end{proposition}
\begin{example}
\end{example}
\begin{example} The resolution $\pi$ of Example \ref{extriplepole} consist of $\alpha=15$ blow-ups and the associated stratification has 35 strata. According to Proposition \ref{formestrpi} we have
$$\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,$$
where $S$ is the subset indexing the 0-dimensional strata of Example \ref{extriplepole} and we have omitted the symbols of differential forms associated to each strata for shortness sake.
\end{example}
@ -1028,7 +1041,7 @@ or equivalently $J_{\check \tau_0}-\frac{(3, m_3)J_{\check \sigma}s}{s+1}+J_{\
\item The dual cone $\check \tau_0$ to the face $\tau_0$ is the cone generated by the rays $$\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)}.$$
\end{enumerate}
Simmilarly we have that
Similarly we have
$$\zeta_f (t) = \frac{(1-t^3)(1-t^{m_3})}{(1-t^{3m_3})^{(m_3,3)}}$$
@ -1073,8 +1086,11 @@ The multiplicity of this cone is $\frac{m_2+2m_3}{(2,m_2)}$. Its contribution t
The multiplicity of this cone is $\frac{2}{(2,m_2)}$. Its contribution to the the local topological zeta function is
$$J_{\check \tau_0}=\frac{2}{\nu_1(3m_2s + 2\nu_1+m_2)}$$
\end{enumerate}
Simmilarly we have that
Similarly we have
$$\zeta_f (t) = \frac{(1-t^3)}{(1-t^{3m_2})^{(m_2,2)}}$$
We remark that this case it is equivalent to $f=z(z^2 - x^{m_2})$.
\subsubsection{Third case $f= z^3 - x^{m_1}z^2 - x^{m_3}$ and $\omega = x^{\nu_1-1}dxdz$ such that $m_3 > m_1$}
@ -1117,6 +1133,12 @@ The multiplicity of this cone is $\frac{2}{(2,m_2)}$. Its contribution to the
$$J_{\check \tau_0}=\frac{m_1}{\nu_1(3m_1s + \nu_1+m_1)}$$
\end{enumerate}
Similarly we have
$$\zeta_f (t) = $$
We remark that this case is equivalent to $f=(z-x^{m_1})(z^2 - x^{m_3-m_2})$
\subsection{Trinomial forms for strata $E^\circ_{i,j}$}
@ -1199,7 +1221,256 @@ $$\frac{1}{\nu_1\nu_2(3s+1)} - \frac{2}{(3m_2s + 2\nu_1+m_2)(3n_2s + 2\nu_2+m_2)
\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 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.
\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
$$Z(F_9, \omega_9)_{o_9}=Z(F_{9,31}, \omega_{9,31})_{o_{9,31}}$$
$$ Z(F_{15}, \omega_{15})_{o_{15}}=Z(F_{15,34}, \omega_{15,34})_{o_{15,34}}$$
\end{example}
\subsection{Second reduction: Elimination of connected components of exceptional divisors containing a $(-1)$-curve}
After performing the First reduction, connected components of exceptional divisors
\begin{proposition}\label{red2}
\end{proposition}
\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}

37
biblio.bib
View File

@ -1,3 +1,40 @@
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Melle Hern\'andez, A.},
TITLE = {Monodromy conjecture for some surface singularities},
JOURNAL = {Ann. Sci. \'Ecole Norm. Sup. (4)},
FJOURNAL = {Annales Scientifiques de l'\'Ecole Normale Sup\'erieure.
Quatri\`eme S\'erie},
VOLUME = {35},
YEAR = {2002},
NUMBER = {4},
PAGES = {605--640},
ISSN = {0012-9593},
MRCLASS = {32S40 (14B05 14G10 32S25)},
MRNUMBER = {1981174},
MRREVIEWER = {Jan\ Stevens},
DOI = {10.1016/S0012-9593(02)01100-X},
URL = {https://doi.org/10.1016/S0012-9593(02)01100-X},
}
@article {GSLM97,
AUTHOR = {Gusein-Zade, S. M. and Luengo, I. and Melle-Hern\'andez, A.},
TITLE = {Partial resolutions and the zeta-function of a singularity},
JOURNAL = {Comment. Math. Helv.},
FJOURNAL = {Commentarii Mathematici Helvetici},
VOLUME = {72},
YEAR = {1997},
NUMBER = {2},
PAGES = {244--256},
ISSN = {0010-2571,1420-8946},
MRCLASS = {32S25 (14B05)},
MRNUMBER = {1470090},
MRREVIEWER = {Jan\ Stevens},
DOI = {10.1007/s000140050014},
URL = {https://doi.org/10.1007/s000140050014},
}
@inproceedings {st:77,
AUTHOR = {Steenbrink, J.H.M.},
TITLE = {Mixed {H}odge structure on the vanishing cohomology},