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\begin{document}
\title{On germs of hypersurfaces with multiplicity 3}
\author{EAB, MGV, ELC}
\maketitle
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 polynomial $F = z^3 + a_1z + a_2z + a_3 \in \CC[[{\bf x}]][z]$ of degree 3 with respect to $z$. By means of a Tschirnhaus transformation, reduce $F$ to a monic equation $z^3 + b_2z + b_3$ that does not have terms of degree 2 with respect to $z$. Secondly, consider an embedded resolution $\pi: X \rightarrow \CC^d$ of the reduced divisor given by the product $b_2b_3$ of the coefficients of degree one and zero with respect to $z$. Consider the usual stratification of the pair $(X, (\pi \circ F)^{-1} \{F=0\})$ defined in terms of decomposition $\cup_{j \in J} E_j$ of $(\pi \circ F)^{-1} \{F=0\}$ into irreducible components. Pulling back $F$ under the map $\pi \times id$, we obtain for each stratum $E_I^\circ$ a quasi-trinomial local expression for $F$.
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.
\section{The quasi-trinomial case}
In this section we consider the case of
$$F = z^3 + M_1(c_1+A)z + M_2(c_2 + M_2) \in \CC[[{\bf x}]][z]$$
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 $M_1 \nmid M_2$.
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);
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\filldraw[black] (-2,-2) circle (2pt) node[anchor=west]{};
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\draw[gray, thick] (6,0,4) -- (13,0,4);
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\draw[gray, thick] (6,3,0) -- (9,1,6);
\draw[gray, dashed] (9,0,0) -- (9,0,6);
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\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.
\
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\draw[gray, thick] (6,0) -- (11,0);
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\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.
\
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\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}
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\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);
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\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 $3c_1 \ne 2c_2$.
\begin{tikzpicture}
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\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}
\bibliographystyle{amsplain}
\bibliography{biblio}
\end{document}

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{
"cells": [
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"S.<x, y, z> = PolynomialRing(QQ, order='neglex')\n",
"f=((y^2+x^3)^2 + x^5*y)*((x^2+y^3)^2 + y^5*x)\n",
"g=(y^6+x^7)*(x^6+y^7)\n",
"h = f^3 - g^2\n",
"F= z^3 - 3 * f * z + 2 * g"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"0"
]
},
"execution_count": 4,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"h = f^3 - g^2\n",
"F.discriminant(z) - 108 * h"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"False"
]
},
"execution_count": 5,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"D=F.discriminant(z)\n",
"load('ZetaFunctionsNewtonND.py')\n",
"ZD = ZetaFunctions(D)\n",
"ZD.is_newton_degenerated(local=True)"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Newton's polyhedron of -108*y^26 + 108*y^30 + 324*x*y^29 + 648*x^2*y^27 + 324*x^2*y^28 + 1296*x^3*y^26 + 108*x^3*y^27 + 648*x^3*y^28 + 1620*x^4*y^24 + 648*x^4*y^25 + 1944*x^4*y^27 + 1944*x^5*y^23 + 3888*x^5*y^25 + 1944*x^5*y^26 + 324*x^5*y^27 - 216*x^6*y^19 + 2160*x^6*y^21 + 324*x^6*y^22 + 7776*x^6*y^24 + 648*x^6*y^25 + 2592*x^6*y^26 + 1080*x^7*y^20 + 9720*x^7*y^22 + 3888*x^7*y^23 + 1944*x^7*y^24 + 5832*x^7*y^25 + 1620*x^8*y^18 + 11664*x^8*y^21 + 13608*x^8*y^23 + 5184*x^8*y^24 + 1296*x^8*y^25 + 324*x^9*y^17 + 12960*x^9*y^19 + 1944*x^9*y^20 + 4860*x^9*y^21 + 21384*x^9*y^22 + 1620*x^9*y^23 + 6048*x^9*y^24 + 648*x^10*y^15 + 7776*x^10*y^18 + 30132*x^10*y^20 + 9720*x^10*y^21 + 7776*x^10*y^22 + 10368*x^10*y^23 + 324*x^10*y^24 + 9720*x^11*y^16 + 6480*x^11*y^18 + 30132*x^11*y^19 + 28512*x^11*y^21 + 7776*x^11*y^22 + 2916*x^11*y^23 + 1944*x^12*y^15 + 36288*x^12*y^17 + 4860*x^12*y^18 + 19440*x^12*y^19 + 33696*x^12*y^20 + 4104*x^12*y^21 + 8424*x^12*y^22 + 3456*x^13*y^13 + 4860*x^13*y^15 + 19440*x^13*y^16 + 55728*x^13*y^18 + 12960*x^13*y^19 + 15552*x^13*y^20 + 11016*x^13*y^21 + 648*x^13*y^22 + 25164*x^14*y^14 + 25920*x^14*y^16 + 42768*x^14*y^17 + 4860*x^14*y^18 + 34992*x^14*y^19 + 6804*x^14*y^20 + 3240*x^14*y^21 + 648*x^15*y^10 + 1944*x^15*y^12 + 4860*x^15*y^13 + 58752*x^15*y^15 + 6480*x^15*y^16 + 34992*x^15*y^17 + 31104*x^15*y^18 + 5508*x^15*y^19 + 6480*x^15*y^20 + 108*x^15*y^21 + 9720*x^16*y^11 + 19440*x^16*y^13 + 25920*x^16*y^14 + 6480*x^16*y^15 + 60264*x^16*y^16 + 9720*x^16*y^17 + 15552*x^16*y^18 + 6480*x^16*y^19 + 648*x^16*y^20 + 324*x^17*y^9 + 36288*x^17*y^12 + 42768*x^17*y^14 + 34992*x^17*y^15 + 9720*x^17*y^16 + 23328*x^17*y^17 + 3888*x^17*y^18 + 1620*x^17*y^19 + 1620*x^18*y^8 + 7776*x^18*y^10 + 6480*x^18*y^11 + 4860*x^18*y^12 + 55728*x^18*y^13 + 4860*x^18*y^14 + 31104*x^18*y^15 + 15552*x^18*y^16 + 3888*x^18*y^17 + 2160*x^18*y^18 - 216*x^19*y^6 + 12960*x^19*y^9 + 30132*x^19*y^11 + 19440*x^19*y^12 + 12960*x^19*y^13 + 34992*x^19*y^14 + 5508*x^19*y^15 + 6480*x^19*y^16 + 1620*x^19*y^17 + 1080*x^20*y^7 + 1944*x^20*y^9 + 30132*x^20*y^10 + 33696*x^20*y^12 + 15552*x^20*y^13 + 6804*x^20*y^14 + 6480*x^20*y^15 + 648*x^20*y^16 + 2160*x^21*y^6 + 11664*x^21*y^8 + 4860*x^21*y^9 + 9720*x^21*y^10 + 28512*x^21*y^11 + 4104*x^21*y^12 + 11016*x^21*y^13 + 3240*x^21*y^14 + 108*x^21*y^15 + 324*x^22*y^6 + 9720*x^22*y^7 + 21384*x^22*y^9 + 7776*x^22*y^10 + 7776*x^22*y^11 + 8424*x^22*y^12 + 648*x^22*y^13 + 1944*x^23*y^5 + 3888*x^23*y^7 + 13608*x^23*y^8 + 1620*x^23*y^9 + 10368*x^23*y^10 + 2916*x^23*y^11 + 1620*x^24*y^4 + 7776*x^24*y^6 + 1944*x^24*y^7 + 5184*x^24*y^8 + 6048*x^24*y^9 + 324*x^24*y^10 + 648*x^25*y^4 + 3888*x^25*y^5 + 648*x^25*y^6 + 5832*x^25*y^7 + 1296*x^25*y^8 - 108*x^26 + 1296*x^26*y^3 + 1944*x^26*y^5 + 2592*x^26*y^6 + 648*x^27*y^2 + 108*x^27*y^3 + 1944*x^27*y^4 + 324*x^27*y^5 + 324*x^28*y^2 + 648*x^28*y^3 + 324*x^29*y + 108*x^30:\n",
" support points = [(0, 26, 0), (0, 30, 0), (1, 29, 0), (2, 27, 0), (2, 28, 0), (3, 26, 0), (3, 27, 0), (3, 28, 0), (4, 24, 0), (4, 25, 0), (4, 27, 0), (5, 23, 0), (5, 25, 0), (5, 26, 0), (5, 27, 0), (6, 19, 0), (6, 21, 0), (6, 22, 0), (6, 24, 0), (6, 25, 0), (6, 26, 0), (7, 20, 0), (7, 22, 0), (7, 23, 0), (7, 24, 0), (7, 25, 0), (8, 18, 0), (8, 21, 0), (8, 23, 0), (8, 24, 0), (8, 25, 0), (9, 17, 0), (9, 19, 0), (9, 20, 0), (9, 21, 0), (9, 22, 0), (9, 23, 0), (9, 24, 0), (10, 15, 0), (10, 18, 0), (10, 20, 0), (10, 21, 0), (10, 22, 0), (10, 23, 0), (10, 24, 0), (11, 16, 0), (11, 18, 0), (11, 19, 0), (11, 21, 0), (11, 22, 0), (11, 23, 0), (12, 15, 0), (12, 17, 0), (12, 18, 0), (12, 19, 0), (12, 20, 0), (12, 21, 0), (12, 22, 0), (13, 13, 0), (13, 15, 0), (13, 16, 0), (13, 18, 0), (13, 19, 0), (13, 20, 0), (13, 21, 0), (13, 22, 0), (14, 14, 0), (14, 16, 0), (14, 17, 0), (14, 18, 0), (14, 19, 0), (14, 20, 0), (14, 21, 0), (15, 10, 0), (15, 12, 0), (15, 13, 0), (15, 15, 0), (15, 16, 0), (15, 17, 0), (15, 18, 0), (15, 19, 0), (15, 20, 0), (15, 21, 0), (16, 11, 0), (16, 13, 0), (16, 14, 0), (16, 15, 0), (16, 16, 0), (16, 17, 0), (16, 18, 0), (16, 19, 0), (16, 20, 0), (17, 9, 0), (17, 12, 0), (17, 14, 0), (17, 15, 0), (17, 16, 0), (17, 17, 0), (17, 18, 0), (17, 19, 0), (18, 8, 0), (18, 10, 0), (18, 11, 0), (18, 12, 0), (18, 13, 0), (18, 14, 0), (18, 15, 0), (18, 16, 0), (18, 17, 0), (18, 18, 0), (19, 6, 0), (19, 9, 0), (19, 11, 0), (19, 12, 0), (19, 13, 0), (19, 14, 0), (19, 15, 0), (19, 16, 0), (19, 17, 0), (20, 7, 0), (20, 9, 0), (20, 10, 0), (20, 12, 0), (20, 13, 0), (20, 14, 0), (20, 15, 0), (20, 16, 0), (21, 6, 0), (21, 8, 0), (21, 9, 0), (21, 10, 0), (21, 11, 0), (21, 12, 0), (21, 13, 0), (21, 14, 0), (21, 15, 0), (22, 6, 0), (22, 7, 0), (22, 9, 0), (22, 10, 0), (22, 11, 0), (22, 12, 0), (22, 13, 0), (23, 5, 0), (23, 7, 0), (23, 8, 0), (23, 9, 0), (23, 10, 0), (23, 11, 0), (24, 4, 0), (24, 6, 0), (24, 7, 0), (24, 8, 0), (24, 9, 0), (24, 10, 0), (25, 4, 0), (25, 5, 0), (25, 6, 0), (25, 7, 0), (25, 8, 0), (26, 0, 0), (26, 3, 0), (26, 5, 0), (26, 6, 0), (27, 2, 0), (27, 3, 0), (27, 4, 0), (27, 5, 0), (28, 2, 0), (28, 3, 0), (29, 1, 0), (30, 0, 0)]\n",
" vertices = [(0, 26, 0), (26, 0, 0), (19, 6, 0), (6, 19, 0)]\n",
" number of proper faces = 19\n",
" Facet 0: -156 + 7*y + 6*x >= 0\n",
" Facet 1: -25 + y + x >= 0\n",
" Facet 2: -156 + 6*y + 7*x >= 0\n",
" Facet 3: x >= 0\n",
" Facet 4: y >= 0\n",
" Facet 5: z >= 0\n"
]
}
],
"source": [
"ZD.give_info_newton(faces=False, cones=False, compact = False)"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"def estrato(ff, chi, pesos):\n",
" Z1 = ZetaFunctions(ff)\n",
" zetamon1 = Z1.monodromy_zeta().value()^chi\n",
" zetatop1 = chi * Z1.topological_zeta(local=True, weights=pesos).value()\n",
" return zetamon1, zetatop1"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Veamos que la resolución larga no aporta\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"(0, 0)"
]
},
"execution_count": 8,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"M3, Z3 = estrato(z^3+x^26,-1, [5,1,1])\n",
"M9, Z9 = estrato(z^3+x^52,-1, [11,1,1])\n",
"M8, Z8 = estrato(z^3+x^26,1, [6,1,1])\n",
"M39, Z39 = estrato(z^3+x^26*y^52,1, [5,11,1])\n",
"M98, Z98 = estrato(z^3+x^52*y^26, 1, [11,6,1])\n",
"M9f, Z9f = estrato(z^3+x^50*y*z + x^52, 1, [11,1,1])\n",
"Z3 + Z39 + Z98 + Z8, Z9 + Z9f\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"Cálculo completo\n",
"\n"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {
"collapsed": false
},
"outputs": [
{
"ename": "SyntaxError",
"evalue": "invalid syntax (3357930205.py, line 3)",
"output_type": "error",
"traceback": [
"\u001b[0;36m Cell \u001b[0;32mIn [9], line 3\u001b[0;36m\u001b[0m\n\u001b[0;31m M17, Z17 = estrato(z**Integer(3)+, Integer(2), [Integer(2),Integer(13),Integer(1)])\u001b[0m\n\u001b[0m ^\u001b[0m\n\u001b[0;31mSyntaxError\u001b[0m\u001b[0;31m:\u001b[0m invalid syntax\n"
]
}
],
"source": [
"M1, Zt1 = estrato(z^3+x^4*z^2 + x^13, -13, [2,1,1])\n",
"M1d, Zt1d = estrato(z^3+x^4*z^2 + x^13*y, 13, [2,1,1])\n",
"M17, Z17 = estrato(z^3+x^4*y^28*z^2 + x^13*y^78 , 2, [2,13,1])\n",
"M7, Z7 = estrato(z^3+x^4*z + x^13*y, -4, [13,1,1])\n",
"M7d, Z7d = estrato(z^3+x^4*z + x^13*y, -4, [13,1,1])\n",
"M7g, Z7g\n"
]
},
{
"cell_type": "code",
"execution_count": 55,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Newton's polyhedron of z^3 + x^4*z + x^13:\n",
" support points = [(0, 0, 3), (4, 0, 1), (13, 0, 0)]\n",
" vertices = [(0, 0, 3), (4, 0, 1), (13, 0, 0)]\n",
" number of proper faces = 15\n",
" Facet 0: z >= 0\n",
" Facet 1: -13 + 9*z + x >= 0\n",
" Facet 2: -6 + 2*z + x >= 0\n",
" Facet 3: x >= 0\n",
" Facet 4: y >= 0\n"
]
},
{
"data": {
"text/plain": [
"(-1/6) * (s + 2/3)^-1 * (s + 1)^-1 * (s - 2)"
]
},
"execution_count": 55,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"G=z^3+x^4*z + x^13\n",
"ZG=ZetaFunctions(G)\n",
"ZG.give_info_newton(faces=False, cones=False, compact = False)\n",
"ZG.topological_zeta(local= True, weights = [2,1,1])"
]
},
{
"cell_type": "code",
"execution_count": 0,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
]
},
{
"cell_type": "code",
"execution_count": 0,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
]
}
],
"metadata": {
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"sage-10.0",
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"display_name": "SageMath 10.0",
"env": {
},
"language": "sagemath",
"metadata": {
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"description": "Open-source mathematical software system",
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},
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