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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 +1208,236 @@ $$\frac{1}{\nu_1\nu_2(3s+1)} - \frac{2}{(3m_2s + 2\nu_1+m_2)(3n_2s + 2\nu_2+m_2) \subsection{} Third case $f= z^3 - x^{m_1}y^{n_1}z^2 - x^{m_3}y^{n_3}$ and such that $3m_1 \leq m_3, 3 n_1 \leq n_3$ and $3(m_1, n_1) \ne 2(m_3, n_3)$. + + + \section{On generic projections} + +% In general, the $z$-axis is not a generic direction of projection for the embedded topological class of the surface given by the Weierstrass polynomial $F=z^3-2f(x,y)z+3g(x,y)$. Therefore the discriminant of $F$ with respect to $z$ given by $f^2+g^3$ is not generic inside the same equivalence class. Let us describe a situation where it is possible to describe a generic topological deformation $F_t=z^3-2f_t(x,y)z+3g_t(x,y)$ of $F$ whose discriminant $f_t^3+g_t^2$ is generic too. + + +%\begin{proposition} +% Assume that the functions $f$ and $g$ admit factorizations of the form $f= x^{2a} f_1$ and $g=x^{3a}g_1$. Moreover, assume that the the multiplicity of the polynomial $F_1(x, y, z)= F(x, y, x^az_1)/x^{3a}$ at the origin is equal or less than $2$. Then, any pair of deformations $f_t$ and $g_t$ of the functions $f$ and $g$ gives a topological deformation $F_t$ of $F$ and a topological deformation $f_t^3+g_t^2$ of $f_t^3+g_t^2$. Moreover, if $f_t^3+g_t^2$ is generic, then $F_t$ is generic. + %\end{proposition} + + \begin{example} Consider the Weierstrass polynomial $F=z^3-3f(x,y)z+2g(x,y)$ with $f=y^2 - x^7 + x^5y$ and $g= f^2 + (y + x^2)^6$. It defines a germ of isolated surface singularity with Milnor number $\mu=41$. The discriminant of $F$ is given by + $$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}