From b844bb5faa7e2a1b099f0df7b04e3c0fb59e3bbb Mon Sep 17 00:00:00 2001 From: "m.gonzalez" Date: Thu, 8 Feb 2024 17:13:31 +0100 Subject: [PATCH 1/6] actualizar Mult3.tex --- Mult3.tex | 641 ++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 641 insertions(+) create mode 100644 Mult3.tex diff --git a/Mult3.tex b/Mult3.tex new file mode 100644 index 0000000..d6ca610 --- /dev/null +++ b/Mult3.tex @@ -0,0 +1,641 @@ +\documentclass[12pt]{amsart} +\usepackage[colorlinks=true,pagebackref,hyperindex,citecolor=Green,linkcolor=Blue]{hyperref} +\usepackage[top=1in, bottom=1in, left=1.1in, right=1.1in]{geometry} +%\usepackage{stmaryrd} +%\usepackage[firstpage]{draftwatermark} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{amsthm} +\usepackage[all]{xy} +\usepackage{mathrsfs} +\usepackage{enumitem} +\usepackage[T1]{fontenc} +\usepackage{color} +\usepackage{stmaryrd} +\usepackage{bigints} + +\usepackage{graphicx} %This is for inserting figures and captions +\usepackage{tikz-cd} %Easy commutative diagrams +\usepackage{tikz} %For plotting sutff, drawin figures... + + +\makeatletter +\def\namedlabel#1#2{\begingroup +#2% +\def\@currentlabel{#2}% +\phantomsection\label{#1}\endgroup +} +\makeatother + +%\begin{enumerate}[noitemsep] + + +\newcommand{\Cech}{ \check{\rm{C}}} +\DeclareMathOperator{\supp}{Supp} +% THEOREM Environments --------------------------------------------------- +\newtheorem{theorem}{Theorem}[section] +\newtheorem*{MainTheorem}{Main Theorem} +\newtheorem{hypothesis}[theorem]{Hypothesis} +\newtheorem{ThmDef}[theorem]{Theorem/Definition} +\newtheorem{corollary}[theorem]{Corollary} +\newtheorem{conjecture}[theorem]{Conjecture} +\newtheorem{lemma}[theorem]{Lemma} +\newcommand{\Fdif}[1]{{ \llbracket #1 \rrbracket }} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{claim}[theorem]{Claim} +\newtheorem{notation}[theorem]{Notation} +\newtheorem*{convention}{Convention} +\newtheorem*{notationCounter}{Notation for Section \ref{counterex}} +\newtheorem*{notationInj}{Notation for Section \ref{Ln&id}} + +\newtheorem{theoremx}{Theorem} +\renewcommand{\thetheoremx}{\Alph{theoremx}} % "letter-numbered" theorems for Introduction + +\newtheorem{propositionx}{Proposition} +\renewcommand{\thepropositionx}{\Alph{propositionx}} % "letter-numbered" theorems for Introduction + + +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{question}[theorem]{Question} +\newtheorem{example}[theorem]{Example} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\numberwithin{equation}{subsection} + +\newtheorem{questionx}[theoremx]{Question} +\renewcommand{\thequestionx}{\Alph{questionx}} % "letter-numbered" questions for Introduction + + +% MATH ------------------------------------------------------------------- + +%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%% + +% EDITING ------------------------------------------------------------------- + +% For author editorial notes 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r@tocindent\number#1\endcsname\relax + }{% + \@tempdima#4\relax + }% + \parindent\z@ \leftskip#3\relax \advance\leftskip\@tempdima\relax + \rightskip\@pnumwidth plus4em \parfillskip-\@pnumwidth + #5\leavevmode\hskip-\@tempdima + \ifcase #1 + \or\or \hskip 1.9em \or \hskip 2em \else \hskip 3em \fi% + #6\nobreak\relax + \dotfill\hbox to\@pnumwidth{\@tocpagenum{#7}}\par + \nobreak + \endgroup + \fi} +\makeatother + +%%%%%%%%%%%%%%%%%%%%% + +% For sets +\newcommand{\set}[1]{\left\{ #1 \right\}} +\renewcommand{\(}{\left(} +\renewcommand{\)}{\right)} +\newcommand{\NN}{\mathbb{N}} +\newcommand{\RR}{\mathbb{R}} +\newcommand{\ZZ}{\mathbb{Z}} +\newcommand{\QQ}{\mathbb{Q}} +\newcommand{\CC}{\mathbb{C}} +\newcommand{\KK}{\mathbb{K}} + +\newcommand{\dif}[1]{^{{\langle #1\rangle}}} + +\newcommand{\m}{\mathfrak{m}} +\newcommand{\n}{\mathfrak{n}} + +%%%%%%%%%%%%%%%%%%%%% + +% For fancy letter +\newcommand{\cM}{\mathscr{M}} +\newcommand{\cN}{\mathscr{N}} +\newcommand{\cA}{\mathscr{A}} +\newcommand{\cK}{K} +\newcommand{\cP}{{\mathcal P}} +\newcommand{\cR}{{\mathcal R}} +\newcommand{\cO}{{\mathcal O}} +\newcommand{\cC}{{\mathcal C}} +\newcommand{\cJ}{{\mathcal J}} +\newcommand{\ff}{\boldsymbol{f}} + +\newcommand{\cB}{\mathcal{B}} + +%%%%%%%%%%%%%%%%%%%%% +% For operators using abbreviations +\newcommand{\Spec}{\operatorname{Spec}} +\newcommand{\Hom}{\operatorname{Hom}} +\newcommand{\sdim}{\operatorname{sdim}} +\newcommand{\Ext}{\operatorname{Ext}} +\newcommand{\Frac}{\operatorname{Frac}} +\newcommand{\InjDim}{\operatorname{inj.dim}} +\newcommand{\Supp}{\operatorname{Supp}} +\newcommand{\Rank}{\operatorname{rank}} +\newcommand{\End}{\operatorname{End}} +\newcommand{\Ker}{\operatorname{Ker}} +\newcommand{\fr}{\operatorname{free.rank}} +\newcommand{\CoKer}{\operatorname{Coker}} +\newcommand{\Ass}{\operatorname{Ass}} +\newcommand{\Ht}{\operatorname{ht}} +\newcommand{\IM}{\operatorname{Im}} +\newcommand{\id}{\operatorname{id}} +\newcommand{\lct}{\operatorname{lct}} + + 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+\newcommand{\ann}{\operatorname{ann}} +\newcommand{\ck}[1]{{#1}^{\vee}} +\newcommand{\Q}{\mathbb{Q}} +\newcommand{\ehk}{\operatorname{e_{HK}}} +\newcommand{\ps}[1]{\llbracket {#1} \rrbracket} +\newcommand{\cd}{\operatorname{cd}} +\newcommand{\Nash}{\operatorname{Nash}} +\renewcommand{\a}{\mathfrak{a}} +\renewcommand{\b}{{J}} +\renewcommand{\b}{{J}} +\newcommand{\OO}{\mathcal{O}} +\newcommand{\JJ}{\mathcal{J}} +%\renewcommand{\ll}{\operatorname{\ell\ell}} +\newcommand{\ctr}{N} % For a generic number of items + +%%%%%%%%%%%%%%%%%%%%% + +% Daniel's commands +\newcommand{\Pnr}{\mathcal{P}^n_R} +\newcommand{\Pn}{\mathcal{P}^n_X} +\newcommand{\OXn}{\Omega^{(n)}_{X/\KK}} +\newcommand{\OXR}{\Omega^{(n)}_{R/\KK}} +\newcommand{\Grr}{\operatorname{Grass}} +\newcommand{\Gr}{\Grr_{\binom{n+d}{d}}(\mathcal{P}^n_X)} +\newcommand{\Ox}{\mathcal O} +\newcommand{\Oxx}{{\mathcal O}_{X,x}} +\newcommand{\Su}{\KK[x^{a_1},\ldots,x^{a_s}]} +\newcommand{\Po}{\KK[x_1,\ldots,x_d]} +\newcommand{\Na}{Nash_n(X)} +\newcommand{\Nn}{\overline{\Nash_n(X)}} +\newcommand{\I}{{\mathcal I}_X} +\newcommand{\V}{\mathbf{V}} + +\newcommand{\Grade}{\operatorname{grade}} +\newcommand{\Pri}{{\mathcal P}^n_X} +\newcommand{\Prix}{{\mathcal P}^n_{\Ox_{X,x}}} +\newcommand{\Prir}{{\mathcal P}^n_{R}} +\newcommand{\Pric}{{\mathcal P}^n_{\Oc}} + + +\newcommand{\T}{\mathbb T} +\newcommand{\A}{{\mathcal A}^Tw} + + +\newcommand{\GrFan}{\operatorname{GF}} +\newcommand{\XGF}{X_{\GrFan(J_n)}} +\newcommand{\GF}{\GrFan(J_n)} +\newcommand{\Nw}{\mathcal{N}_{\sigma}(\mathcal{I})} +\newcommand{\sd}{\check{\sigma}} +\newcommand{\Si}{\mbox{Sing}} +\newcommand{\Co}{\mbox{Conv}} + +\newcommand{\sz}{\sigma_{\ZZ}} +\newcommand{\Suc}{\CC[u,u^3v^4,uv]} +\newcommand{\Suk}{\KK[u,u^3v^4,uv]} +\newcommand{\Suz}{\ZZ[u,u^3v^4,uv]} +\newcommand{\Jn}{\langle u-1,u^3v^4-1,uv-1 \rangle^{n+1}} +\newcommand{\Gn}{\mathbb{G}_n} +\newcommand{\Mn}{\mathbb{M}_n} +\newcommand{\cV}{\mathcal{V}} +\newcommand{\Gc}{\mathbb{G}_n^0} +\newcommand{\Gp}{\mathbb{G}_n^p} +\newcommand{\Pnn}{\mathcal{P}_n} +\newcommand{\Pnu}{\mathcal{P}_{n-1}} +\newcommand{\Cc}{C_{\mathbb{G}_n^{(0)}}} +\newcommand{\Cp}{C_{\mathbb{G}_n^{(p)}}} +\newcommand{\bfg}{{\bf \left(\frac{f}{g}\right)^s}} +\newcommand{\bfga}{{\bf \frac{f^s}{g^{s+\alpha}}}} +\newcommand{\bfgt}{{\bf \frac{\tilde{f}^s}{\tilde{g}^{s+\alpha}}}} +\newcommand{\bfgOne}{{\bf \frac{f^s}{g^{s+1}}}} + +\newcommand{\ga}{{\bf \frac{1}{g^{\alpha}}}} +\newcommand{\gd}{\operatorname{gl.dim}} + +\newcommand{\lm}{\operatorname{lm}} +\newcommand{\lc}{\operatorname{lc}} +\newcommand{\lt}{\operatorname{lt}} +\newcommand{\init}{\operatorname{in}} +\newcommand{\diff}{\operatorname{diff}} + + +%------------------------------------------------------- + +\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); +\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 $3c_1 \ne 2c_2$. + + +\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} + + + + + +\bibliographystyle{amsplain} +\bibliography{biblio} + + + +\end{document} \ No newline at end of file From 1a03efcc8ee06757518b29d6540dd1e5b7cde51a Mon Sep 17 00:00:00 2001 From: Enrique Artal Date: Tue, 13 Feb 2024 09:02:20 +0100 Subject: [PATCH 2/6] cambios antiguos --- Mult3.tex | 628 ++++++++++++++++++++++++ NonisoMHS.pdf | Bin 0 -> 168708 bytes Singmult3Ejem1.ipynb | 1079 ++++++++++++++++++++++++++++++++++++++++++ 3 files changed, 1707 insertions(+) create mode 100644 Mult3.tex create mode 100644 NonisoMHS.pdf create mode 100644 Singmult3Ejem1.ipynb diff --git a/Mult3.tex b/Mult3.tex new file mode 100644 index 0000000..85fe89a --- /dev/null +++ b/Mult3.tex @@ -0,0 +1,628 @@ +\documentclass[12pt]{amsart} +\usepackage[colorlinks=true,pagebackref,hyperindex,citecolor=Green,linkcolor=Blue]{hyperref} +\usepackage[top=1in, bottom=1in, left=1.1in, right=1.1in]{geometry} +%\usepackage{stmaryrd} +%\usepackage[firstpage]{draftwatermark} +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{amsthm} +\usepackage[all]{xy} +\usepackage{mathrsfs} +\usepackage{enumitem} +\usepackage[T1]{fontenc} +\usepackage{color} +\usepackage{stmaryrd} +\usepackage{bigints} + +\usepackage{graphicx} %This is for inserting figures and captions +\usepackage{tikz-cd} %Easy commutative diagrams +\usepackage{tikz} %For plotting sutff, drawin figures... + + +\makeatletter +\def\namedlabel#1#2{\begingroup +#2% +\def\@currentlabel{#2}% +\phantomsection\label{#1}\endgroup +} +\makeatother + +%\begin{enumerate}[noitemsep] + + +\newcommand{\Cech}{ \check{\rm{C}}} +\DeclareMathOperator{\supp}{Supp} +% THEOREM Environments --------------------------------------------------- +\newtheorem{theorem}{Theorem}[section] +\newtheorem*{MainTheorem}{Main Theorem} +\newtheorem{hypothesis}[theorem]{Hypothesis} +\newtheorem{ThmDef}[theorem]{Theorem/Definition} +\newtheorem{corollary}[theorem]{Corollary} +\newtheorem{conjecture}[theorem]{Conjecture} +\newtheorem{lemma}[theorem]{Lemma} +\newcommand{\Fdif}[1]{{ \llbracket #1 \rrbracket }} +\newtheorem{proposition}[theorem]{Proposition} +\newtheorem{claim}[theorem]{Claim} +\newtheorem{notation}[theorem]{Notation} +\newtheorem*{convention}{Convention} +\newtheorem*{notationCounter}{Notation for Section \ref{counterex}} +\newtheorem*{notationInj}{Notation for Section \ref{Ln&id}} + +\newtheorem{theoremx}{Theorem} +\renewcommand{\thetheoremx}{\Alph{theoremx}} % "letter-numbered" theorems for Introduction + +\newtheorem{propositionx}{Proposition} +\renewcommand{\thepropositionx}{\Alph{propositionx}} % "letter-numbered" theorems for Introduction + + +\theoremstyle{definition} +\newtheorem{definition}[theorem]{Definition} +\newtheorem{question}[theorem]{Question} +\newtheorem{example}[theorem]{Example} +\theoremstyle{remark} +\newtheorem{remark}[theorem]{Remark} +\numberwithin{equation}{subsection} + +\newtheorem{questionx}[theoremx]{Question} +\renewcommand{\thequestionx}{\Alph{questionx}} % "letter-numbered" questions for Introduction + + +% MATH ------------------------------------------------------------------- + +%%%%%%%%%%%%%%%%%%%%% +%%%%%%%%%%%%%%%%%%%%% + +% EDITING ------------------------------------------------------------------- + +% For author editorial notes +%\newcommand{\note}[2]{} + \newcommand{\note}[2]{ {\color{blue} + \vspace{2 mm} + \par \noindent + 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\advance\leftskip\@tempdima\relax + \rightskip\@pnumwidth plus4em \parfillskip-\@pnumwidth + #5\leavevmode\hskip-\@tempdima + \ifcase #1 + \or\or \hskip 1.9em \or \hskip 2em \else \hskip 3em \fi% + #6\nobreak\relax + \dotfill\hbox to\@pnumwidth{\@tocpagenum{#7}}\par + \nobreak + \endgroup + \fi} +\makeatother + +%%%%%%%%%%%%%%%%%%%%% + +% For sets +\newcommand{\set}[1]{\left\{ #1 \right\}} +\renewcommand{\(}{\left(} +\renewcommand{\)}{\right)} +\newcommand{\NN}{\mathbb{N}} +\newcommand{\RR}{\mathbb{R}} +\newcommand{\ZZ}{\mathbb{Z}} +\newcommand{\QQ}{\mathbb{Q}} +\newcommand{\CC}{\mathbb{C}} +\newcommand{\KK}{\mathbb{K}} + +\newcommand{\dif}[1]{^{{\langle #1\rangle}}} + +\newcommand{\m}{\mathfrak{m}} +\newcommand{\n}{\mathfrak{n}} + +%%%%%%%%%%%%%%%%%%%%% + +% For fancy letter +\newcommand{\cM}{\mathscr{M}} +\newcommand{\cN}{\mathscr{N}} +\newcommand{\cA}{\mathscr{A}} +\newcommand{\cK}{K} +\newcommand{\cP}{{\mathcal P}} +\newcommand{\cR}{{\mathcal R}} +\newcommand{\cO}{{\mathcal O}} +\newcommand{\cC}{{\mathcal C}} +\newcommand{\cJ}{{\mathcal J}} +\newcommand{\ff}{\boldsymbol{f}} + +\newcommand{\cB}{\mathcal{B}} + +%%%%%%%%%%%%%%%%%%%%% +% For operators using abbreviations +\newcommand{\Spec}{\operatorname{Spec}} +\newcommand{\Hom}{\operatorname{Hom}} +\newcommand{\sdim}{\operatorname{sdim}} +\newcommand{\Ext}{\operatorname{Ext}} +\newcommand{\Frac}{\operatorname{Frac}} +\newcommand{\InjDim}{\operatorname{inj.dim}} +\newcommand{\Supp}{\operatorname{Supp}} +\newcommand{\Rank}{\operatorname{rank}} +\newcommand{\End}{\operatorname{End}} +\newcommand{\Ker}{\operatorname{Ker}} +\newcommand{\fr}{\operatorname{free.rank}} +\newcommand{\CoKer}{\operatorname{Coker}} +\newcommand{\Ass}{\operatorname{Ass}} +\newcommand{\Ht}{\operatorname{ht}} +\newcommand{\IM}{\operatorname{Im}} +\newcommand{\id}{\operatorname{id}} +\newcommand{\lct}{\operatorname{lct}} + + +\newcommand{\Min}{\operatorname{min}} +\newcommand{\fpt}{\operatorname{fpt}} +\newcommand{\gr}{\operatorname{gr}} +\newcommand{\ch}{\operatorname{char}} +\newcommand{\Der}{\operatorname{Der}} +\newcommand{\tors}{\operatorname{tors}} +\newcommand{\re}{\operatorname{re}} + +%%%%%%%%%%%%%%%%%%%%% + +% Maps +\newcommand{\FDer}[1]{\stackrel{#1}{\to}} +\newcommand{\FDerL}[1]{\stackrel{#1}{\longrightarrow }} +\newcommand{\onto}{\twoheadrightarrow} +\newcommand{\connarrow}{\ar `[d] `[l] `[llld] `[ld] [lld]} %Connecting map in LES + +%%%%%%%%%%%%%%%%%%%%% + +% Alessandro's commands +\newcommand{\ls}{\leqslant} +\newcommand{\gs}{\geqslant} +\newcommand{\ds}{\displaystyle} +\newcommand{\eee}[1]{\underline{}^e \hspace{-0.05cm}{#1}} +\newcommand{\eeep}[1]{\underline{}^{e+e'} \hspace{-0.05cm}{#1}} +\newcommand{\p}{\mathfrak{p}} +\newcommand{\q}{\mathfrak{q}} +\newcommand{\MIN}{\operatorname{Min}} +\newcommand{\ord}{\operatorname{ord}} +\newcommand{\ov}[1]{\overline{#1}} +\newcommand{\ann}{\operatorname{ann}} +\newcommand{\ck}[1]{{#1}^{\vee}} +\newcommand{\Q}{\mathbb{Q}} +\newcommand{\ehk}{\operatorname{e_{HK}}} +\newcommand{\ps}[1]{\llbracket {#1} \rrbracket} +\newcommand{\cd}{\operatorname{cd}} +\newcommand{\Nash}{\operatorname{Nash}} +\renewcommand{\a}{\mathfrak{a}} +\renewcommand{\b}{{J}} +\renewcommand{\b}{{J}} +\newcommand{\OO}{\mathcal{O}} +\newcommand{\JJ}{\mathcal{J}} +%\renewcommand{\ll}{\operatorname{\ell\ell}} +\newcommand{\ctr}{N} % For a generic number of items + +%%%%%%%%%%%%%%%%%%%%% + +% Daniel's commands +\newcommand{\Pnr}{\mathcal{P}^n_R} +\newcommand{\Pn}{\mathcal{P}^n_X} +\newcommand{\OXn}{\Omega^{(n)}_{X/\KK}} +\newcommand{\OXR}{\Omega^{(n)}_{R/\KK}} +\newcommand{\Grr}{\operatorname{Grass}} +\newcommand{\Gr}{\Grr_{\binom{n+d}{d}}(\mathcal{P}^n_X)} +\newcommand{\Ox}{\mathcal O} +\newcommand{\Oxx}{{\mathcal O}_{X,x}} +\newcommand{\Su}{\KK[x^{a_1},\ldots,x^{a_s}]} +\newcommand{\Po}{\KK[x_1,\ldots,x_d]} +\newcommand{\Na}{Nash_n(X)} +\newcommand{\Nn}{\overline{\Nash_n(X)}} +\newcommand{\I}{{\mathcal I}_X} +\newcommand{\V}{\mathbf{V}} + +\newcommand{\Grade}{\operatorname{grade}} +\newcommand{\Pri}{{\mathcal P}^n_X} +\newcommand{\Prix}{{\mathcal P}^n_{\Ox_{X,x}}} +\newcommand{\Prir}{{\mathcal P}^n_{R}} +\newcommand{\Pric}{{\mathcal P}^n_{\Oc}} + + +\newcommand{\T}{\mathbb T} +\newcommand{\A}{{\mathcal A}^Tw} + + +\newcommand{\GrFan}{\operatorname{GF}} +\newcommand{\XGF}{X_{\GrFan(J_n)}} +\newcommand{\GF}{\GrFan(J_n)} +\newcommand{\Nw}{\mathcal{N}_{\sigma}(\mathcal{I})} +\newcommand{\sd}{\check{\sigma}} +\newcommand{\Si}{\mbox{Sing}} +\newcommand{\Co}{\mbox{Conv}} + +\newcommand{\sz}{\sigma_{\ZZ}} +\newcommand{\Suc}{\CC[u,u^3v^4,uv]} +\newcommand{\Suk}{\KK[u,u^3v^4,uv]} +\newcommand{\Suz}{\ZZ[u,u^3v^4,uv]} +\newcommand{\Jn}{\langle u-1,u^3v^4-1,uv-1 \rangle^{n+1}} +\newcommand{\Gn}{\mathbb{G}_n} +\newcommand{\Mn}{\mathbb{M}_n} +\newcommand{\cV}{\mathcal{V}} +\newcommand{\Gc}{\mathbb{G}_n^0} +\newcommand{\Gp}{\mathbb{G}_n^p} +\newcommand{\Pnn}{\mathcal{P}_n} +\newcommand{\Pnu}{\mathcal{P}_{n-1}} +\newcommand{\Cc}{C_{\mathbb{G}_n^{(0)}}} +\newcommand{\Cp}{C_{\mathbb{G}_n^{(p)}}} +\newcommand{\bfg}{{\bf \left(\frac{f}{g}\right)^s}} +\newcommand{\bfga}{{\bf \frac{f^s}{g^{s+\alpha}}}} +\newcommand{\bfgt}{{\bf \frac{\tilde{f}^s}{\tilde{g}^{s+\alpha}}}} +\newcommand{\bfgOne}{{\bf \frac{f^s}{g^{s+1}}}} + +\newcommand{\ga}{{\bf \frac{1}{g^{\alpha}}}} +\newcommand{\gd}{\operatorname{gl.dim}} + +\newcommand{\lm}{\operatorname{lm}} +\newcommand{\lc}{\operatorname{lc}} +\newcommand{\lt}{\operatorname{lt}} +\newcommand{\init}{\operatorname{in}} +\newcommand{\diff}{\operatorname{diff}} + + +%------------------------------------------------------- + +\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-trimonial 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 $M2$ 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); +\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]{$m_1$}; + + +\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]{$m_1$}; + + + + + +\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} + + + + + +\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 $3c_1 \ne 2c_2$. + + +\begin{tikzpicture} + + + + + +\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] (6,1,3) -- (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} + + + + + +\bibliographystyle{amsplain} +\bibliography{biblio} + + + +\end{document} \ No newline at end of file diff --git a/NonisoMHS.pdf b/NonisoMHS.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0fba1dff5e2693ffa217d187fd81450594a543aa GIT binary patch literal 168708 zcma&NQ;aT9)2-XK?S9*~wc4v~+qP}nwr$(CZF{xdy}xsEzT_nTzWC==RZ=&VnatF9 zMolt#k-v0I^lUI>^UEXaFigw2(V*3Kr5 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z7TKBz*W!1u{Jk2JKSiDXbNbZZ!4nN7Q!`jbDO(d$H)3rjVrC{bU06nCOApiELs&+2 zVr^z(R$^x0q>`PTGjQ^EBD2!(Jn`tej z)6F;Iatu2cojQLeo($bIr$BN@LvvwqiF#1!iXhJmgok!;h!c6Ai=+L`Wt{JegP5pK zM%cXkkd)ttmB;cb*g(01lZHi%5u60Q`4LmQUi6DAW{a zCW-OiQ6&MEl?kz+Nf3N)VKwUnZ6xa5(yDn4St)2nP#q6{;;8)0OQVf7WUIFLtcYkE-yIqItP;hTFwMD~^wE+< sp2hyro3GU)N#I(X`VTc;oSY3Eo!x-pLRe-F05dxPmYiHnUL5xS0;9xOvj6}9 literal 0 HcmV?d00001 diff --git a/Singmult3Ejem1.ipynb b/Singmult3Ejem1.ipynb new file mode 100644 index 0000000..eed404a --- /dev/null +++ b/Singmult3Ejem1.ipynb @@ -0,0 +1,1079 @@ +{ + "cells": [ + { + "cell_type": "markdown", + "id": "e8d28f5d-556f-4e0d-9ec8-e039ca35cdf4", + "metadata": {}, + "source": [] + }, + { + "cell_type": "code", + "execution_count": 2, + "id": "555b4804-430a-43cd-8c2e-1292899c6ea7", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((-1) * (t - 1) * (t + 1) * (t^4 - t^3 + t^2 - t + 1) * (t^4 + t^3 + t^2 + t + 1) * (t^6 + t^3 + 1),\n", + " (-11/90) * (s + 8/9)^-1 * (s + 9/10)^-1 * (s + 1)^-1 * (s^2 - 69/11*s - 72/11),\n", + " x^5*y^2 + x^2*y^4)" + ] + }, + "execution_count": 2, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "load('../Descargas/ZetaFunctionsNewtonND.py')\n", + "R. = QQ[]\n", + "F = z^3+(y^2+x^3)*z+x^2*y^2\n", + "Z = ZetaFunctions(F)\n", + "Zmon=Z.monodromy_zeta()\n", + "Ztop=Z.topological_zeta(local = True)\n", + "g=(y^2+x^3)*x^2*y^2\n", + "Zmon, Ztop, g" + ] + }, + { + "cell_type": "markdown", + "id": "b6ea28e6-3ece-4e52-a34a-e00bdf345007", + "metadata": {}, + "source": [ + "Primera explosión ---> divisor E_1 (6,2)" + ] + }, + { + "cell_type": "code", + "execution_count": 3, + "id": "adf9be35-2eda-463f-9581-0f17b879eacb", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(x^5*y + x^2,\n", + " x^3*y^3*z + x^2*y^4 + y^2*z + z^3,\n", + " y^4 + x*y^2,\n", + " x^4*y^2 + x^2*y^2*z + x^3*z + z^3)" + ] + }, + "execution_count": 3, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "g1=g(x=x*y)/y^6\n", + "F1=F(x=x*y)\n", + "g1b=g(y=x*y)/x^6\n", + "F1b=F(y=x*y)\n", + "g1,F1, g1b, F1b" + ] + }, + { + "cell_type": "markdown", + "id": "1f1c1819-19bc-45c7-8bec-ddd1b9380b55", + "metadata": {}, + "source": [ + "Aportación de estrato de intersección E_1 (6,2) y g_2=x2" + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "id": "fa867c2c-1819-4500-bf4b-fd29acfdeb14", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((-1/2) * (s + 1)^-2 * (s - 1), (-1) * (t - 1)^-1 * (t^2 + t + 1)^-1)" + ] + }, + "execution_count": 26, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Z1 = ZetaFunctions(F1)\n", + "A1=Z1.topological_zeta(local=True, weights=[1,2,1])\n", + "M1=Z1.monodromy_zeta()\n", + "A1, M1" + ] + }, + { + "cell_type": "markdown", + "id": "cbef6f22-df04-4d42-ac7f-87917c7d2619", + "metadata": {}, + "source": [ + "Aportación de estrato abierto de E_1 (6,2)" + ] + }, + { + "cell_type": "code", + "execution_count": 27, + "id": "239b3798-b6a7-43bb-a94c-d29a366d5617", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((-1/2) * (s + 1)^-2 * (s - 1), (-1) * (t - 1)^-1 * (t^2 + t + 1)^-1)" + ] + }, + "execution_count": 27, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "F1t=F1(x=x+2)\n", + "Z1t = ZetaFunctions(F1t)\n", + "A1t=Z1t.topological_zeta(local=True, weights=[1,2,1])\n", + "M1t=Z1t.monodromy_zeta()\n", + "A1t, M1t" + ] + }, + { + "cell_type": "markdown", + "id": "76776c98-c8d4-499a-a1ee-6aca1d182c21", + "metadata": {}, + "source": [ + "Segunda explosión E_2 (9,3)" + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "id": "062e5987-292d-4e31-996a-db366df666b9", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(x + y,\n", + " x^4*y^6 + x^3*y^3*z + x^2*y^4*z + z^3,\n", + " x*y^4 + y^2,\n", + " x^6*y^2 + x^4*y^2*z + x^3*z + z^3)" + ] + }, + "execution_count": 28, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "g2=g1b(x=x*y)/y^3\n", + "F2=F1b(x=x*y)\n", + "g2b=g1b(y=x*y)/x^3\n", + "F2b=F1b(y=x*y)\n", + "g2, F2, g2b, F2b" + ] + }, + { + "cell_type": "markdown", + "id": "40cd5f3c-2358-466a-9107-27b2dfe415a2", + "metadata": {}, + "source": [ + "Aportación de estrato de intersección E_2 (9,3) y g_3=y2" + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "id": "2c6a512e-4d1f-41af-936d-43065a2bef90", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((1/3) * (s + 1)^-2, (t^6 + t^3 + 1)^-1)" + ] + }, + "execution_count": 29, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Z2b = ZetaFunctions(F2b)\n", + "A2b=Z2b.topological_zeta(local=True, weights=[3,1,1])\n", + "M2b=Z2b.monodromy_zeta()\n", + "A2b, M2b" + ] + }, + { + "cell_type": "markdown", + "id": "330f44ff-7833-44c3-a4fa-0e646237ab49", + "metadata": {}, + "source": [ + "Aportación de estrato abierto de E_1 (6,2)" + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "id": "8948fe24-90c2-450c-a6cc-134a2c82d080", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((1/3) * (s + 1)^-2, (t^6 + t^3 + 1)^-1)" + ] + }, + "execution_count": 30, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "F2t=F2(x=x+3)\n", + "Z2t = ZetaFunctions(F2t)\n", + "A2t=Z2t.topological_zeta(local=True, weights=[1,3,1])\n", + "M2t=Z2t.monodromy_zeta()\n", + "A2t, M2t" + ] + }, + { + "cell_type": "markdown", + "id": "2efb4eb5-e6bd-43de-8b96-bf3bce3119f6", + "metadata": {}, + "source": [ + "Tercera explosión E_3 (16,5)" + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "id": "dd7d3923-e444-4f72-8b3b-cd093db11ed5", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(x + 1,\n", + " x^4*y^10 + x^3*y^6*z + x^2*y^6*z + z^3,\n", + " y + 1,\n", + " x^10*y^6 + x^6*y^4*z + x^6*y^3*z + z^3)" + ] + }, + "execution_count": 31, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "g3=g2(x=x*y)/y\n", + "F3=F2(x=x*y)\n", + "g3b=g2(y=x*y)/x\n", + "F3b=F2(y=x*y)\n", + "g3, F3, g3b, F3b" + ] + }, + { + "cell_type": "code", + "execution_count": 32, + "id": "61cd0e41-ae53-4dc2-a822-5992ab34d2e2", + "metadata": {}, + "outputs": [ + { + "ename": "SyntaxError", + "evalue": "invalid syntax (1948788351.py, line 1)", + "output_type": "error", + "traceback": [ + "\u001b[0;36m Cell \u001b[0;32mIn [32], line 1\u001b[0;36m\u001b[0m\n\u001b[0;31m Aportación de intersección de E_1 (Integer(6),Integer(2)) y E_3 (Integer(16),Integer(5))\u001b[0m\n\u001b[0m ^\u001b[0m\n\u001b[0;31mSyntaxError\u001b[0m\u001b[0;31m:\u001b[0m invalid syntax\n" + ] + } + ], + "source": [ + "Aportación de intersección de E_1 (6,2) y E_3 (16,5)" + ] + }, + { + "cell_type": "code", + "execution_count": 33, + "id": "e74156ef-b808-4581-9fe5-72cd237c73c3", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((1/30) * (s + 1)^-2 * (s + 8/9)^-1 * (s^2 + 1/3*s + 8/3),\n", + " (-1) * (t - 1) * (t^2 + t + 1))" + ] + }, + "execution_count": 33, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Z13 = ZetaFunctions(F3)\n", + "A13=Z13.topological_zeta(local=True, weights=[2,5,1])\n", + "M13=Z13.monodromy_zeta()\n", + "A13, M13" + ] + }, + { + "cell_type": "markdown", + "id": "fdc916f8-9529-4a65-b3f5-ca51808d95e3", + "metadata": {}, + "source": [ + "Aportación de intersección de E_2 (9,3) y E_3 (16,5)" + ] + }, + { + "cell_type": "code", + "execution_count": 34, + "id": "699c1fd9-b59c-468d-9e17-63e6b189eb64", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((1/30) * (s + 1)^-2 * (s + 8/9)^-1 * (s^2 + 1/3*s + 8/3),\n", + " (-1) * (t - 1) * (t^2 + t + 1))" + ] + }, + "execution_count": 34, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Z23 = ZetaFunctions(F3b)\n", + "A23=Z23.topological_zeta(local=True, weights=[5,3,1])\n", + "M23=Z23.monodromy_zeta()\n", + "A13, M23" + ] + }, + { + "cell_type": "markdown", + "id": "f8c4cef1-0d85-4dd1-9c84-1feeeebd81c7", + "metadata": {}, + "source": [ + "Aportación de estrato abierto de E_3 (16,5)" + ] + }, + { + "cell_type": "code", + "execution_count": 35, + "id": "253d9222-7bfb-463a-bca6-6aa883f9eda9", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((-7/45) * (s + 8/9)^-1 * (s + 1)^-1 * (s - 8/7),\n", + " (-1) * (t - 1)^-1 * (t^2 + t + 1)^-1 * (t^6 + t^3 + 1)^-2)" + ] + }, + "execution_count": 35, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "F3t=F3(x=x+3)\n", + "Z3t = ZetaFunctions(F3t)\n", + "A3t=Z3t.topological_zeta(local=True, weights=[1,5,1])\n", + "M3t=Z3t.monodromy_zeta()\n", + "A3t, M3t" + ] + }, + { + "cell_type": "code", + "execution_count": 36, + "id": "52ccfc68-0bb7-47ee-b70c-e9e2982bc645", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((-7/45) * (s + 8/9)^-1 * (s + 1)^-1 * (s - 8/7),\n", + " (-1) * (t - 1)^-1 * (t^2 + t + 1)^-1 * (t^6 + t^3 + 1)^-2)" + ] + }, + "execution_count": 36, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "F3bt=F3b(y=y+3)\n", + "Z3bt = ZetaFunctions(F3bt)\n", + "A3bt=Z3bt.topological_zeta(local=True, weights=[5,1,1])\n", + "M3bt=Z3bt.monodromy_zeta()\n", + "A3bt, M3bt" + ] + }, + { + "cell_type": "markdown", + "id": "1fd26ac5-19fd-4dad-9015-87354ac4074a", + "metadata": {}, + "source": [ + "Aportación de intersección de E_3 (16,5) y g_1" + ] + }, + { + "cell_type": "code", + "execution_count": 37, + "id": "ed999fd8-d5fc-4f59-a1b1-a5162637d51b", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((1/6) * (s + 8/9)^-1 * (s + 9/10)^-1 * (s + 1)^-1 * (s^2 + 142/75*s + 24/25),\n", + " (t + 1) * (t - 1)^2 * (t^2 + t + 1) * (t^4 - t^3 + t^2 - t + 1) * (t^4 + t^3 + t^2 + t + 1))" + ] + }, + "execution_count": 37, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "F30=F3(x=x-1)\n", + "Z30 = ZetaFunctions(F30)\n", + "A30=Z30.topological_zeta(local=True, weights=[1,5,1])\n", + "M30=Z30.monodromy_zeta()\n", + "A30, M30" + ] + }, + { + "cell_type": "code", + "execution_count": 38, + "id": "7856f357-7f92-49c5-a87c-0684a24f93fd", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((1/6) * (s + 8/9)^-1 * (s + 9/10)^-1 * (s + 1)^-1 * (s^2 + 142/75*s + 24/25),\n", + " (t + 1) * (t - 1)^2 * (t^2 + t + 1) * (t^4 - t^3 + t^2 - t + 1) * (t^4 + t^3 + t^2 + t + 1))" + ] + }, + "execution_count": 38, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "F3b0=F3b(y=y-1)\n", + "Z3b0 = ZetaFunctions(F3b0)\n", + "A3b0=Z3b0.topological_zeta(local=True, weights=[5,1,1])\n", + "M3b0=Z3b0.monodromy_zeta()\n", + "A3b0, M3b0" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "dc98fc4f-cb55-4c1b-ab70-74da3356a3d5", + "metadata": {}, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "2214bc22-c6f8-4cf1-b7b3-7ef60d1bf5e5", + "metadata": {}, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": 49, + "id": "9328b08b-cc6a-42fa-818f-c9c6b237f40b", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0" + ] + }, + "execution_count": 49, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "T. = QQ[]\n", + "A=A1+A2b+A13+A23-A3t+A30\n", + "B=(-11/90) * (s + 8/9)^-1 * (s + 9/10)^-1 * (s + 1)^-1 * (s^2 - 69/11*s - 72/11)\n", + "A-B" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "95641a8c-4454-45fa-8353-9278d5937ee3", + "metadata": {}, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "7eea46a4-1439-40de-a16c-be14ce2de2fa", + "metadata": {}, + "outputs": [], + "source": [] + }, + { + "cell_type": "markdown", + "id": "2bb58afd", + "metadata": {}, + "source": [] + }, + { + "cell_type": "code", + "execution_count": 24, + "id": "823aaa80", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((-1) * (t - 1) * (t + 1) * (t^4 - t^3 + t^2 - t + 1) * (t^4 + t^3 + t^2 + t + 1) * (t^6 + t^3 + 1),\n", + " (-11/90) * (s + 8/9)^-1 * (s + 9/10)^-1 * (s + 1)^-1 * (s^2 - 69/11*s - 72/11),\n", + " x^5*y^2 + x^2*y^4)" + ] + }, + "execution_count": 24, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "load('ZetaFunctionsNewtonND.py')\n", + "R. = QQ[]\n", + "F = z^3+(y^2+x^3)*z+x^2*y^2\n", + "Z = ZetaFunctions(F)\n", + "Zmon=Z.monodromy_zeta()\n", + "Ztop=Z.topological_zeta(local = True)\n", + "g=(y^2+x^3)*x^2*y^2\n", + "Zmon, Ztop, g" + ] + }, + { + "cell_type": "markdown", + "id": "70c3cf27", + "metadata": {}, + "source": [ + "Primera explosión ---> divisor E_1 (6,2)" + ] + }, + { + "cell_type": "code", + "execution_count": 25, + "id": "b68f383a", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(x^5*y + x^2,\n", + " x^3*y^3*z + x^2*y^4 + y^2*z + z^3,\n", + " y^4 + x*y^2,\n", + " x^4*y^2 + x^2*y^2*z + x^3*z + z^3)" + ] + }, + "execution_count": 25, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "g1=g(x=x*y)/y^6\n", + "F1=F(x=x*y)\n", + "g1b=g(y=x*y)/x^6\n", + "F1b=F(y=x*y)\n", + "g1,F1, g1b, F1b" + ] + }, + { + "cell_type": "markdown", + "id": "aacd2b71", + "metadata": {}, + "source": [ + "Aportación de estrato de intersección E_1 (6,2) y g_2=x2" + ] + }, + { + "cell_type": "code", + "execution_count": 26, + "id": "80675fc5", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((-1/2) * (s + 1)^-2 * (s - 1), (-1) * (t - 1)^-1 * (t^2 + t + 1)^-1)" + ] + }, + "execution_count": 26, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Z1 = ZetaFunctions(F1)\n", + "A1=Z1.topological_zeta(local=True, weights=[1,2,1])\n", + "M1=Z1.monodromy_zeta()\n", + "A1, M1" + ] + }, + { + "cell_type": "markdown", + "id": "724f3e35", + "metadata": {}, + "source": [ + "Aportación de estrato abierto de E_1 (6,2)" + ] + }, + { + "cell_type": "code", + "execution_count": 27, + "id": "c78f4bf8", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((-1/2) * (s + 1)^-2 * (s - 1), (-1) * (t - 1)^-1 * (t^2 + t + 1)^-1)" + ] + }, + "execution_count": 27, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "F1t=F1(x=x+2)\n", + "Z1t = ZetaFunctions(F1t)\n", + "A1t=Z1t.topological_zeta(local=True, weights=[1,2,1])\n", + "M1t=Z1t.monodromy_zeta()\n", + "A1t, M1t" + ] + }, + { + "cell_type": "markdown", + "id": "d5b62001", + "metadata": {}, + "source": [ + "Segunda explosión E_2 (9,3)" + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "id": "5918cb49", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(x + y,\n", + " x^4*y^6 + x^3*y^3*z + x^2*y^4*z + z^3,\n", + " x*y^4 + y^2,\n", + " x^6*y^2 + x^4*y^2*z + x^3*z + z^3)" + ] + }, + "execution_count": 28, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "g2=g1b(x=x*y)/y^3\n", + "F2=F1b(x=x*y)\n", + "g2b=g1b(y=x*y)/x^3\n", + "F2b=F1b(y=x*y)\n", + "g2, F2, g2b, F2b" + ] + }, + { + "cell_type": "markdown", + "id": "d60dcc57", + "metadata": {}, + "source": [ + "Aportación de estrato de intersección E_2 (9,3) y g_3=y2" + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "id": "7c5fb9eb", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((1/3) * (s + 1)^-2, (t^6 + t^3 + 1)^-1)" + ] + }, + "execution_count": 29, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Z2b = ZetaFunctions(F2b)\n", + "A2b=Z2b.topological_zeta(local=True, weights=[3,1,1])\n", + "M2b=Z2b.monodromy_zeta()\n", + "A2b, M2b" + ] + }, + { + "cell_type": "markdown", + "id": "4a56a110", + "metadata": {}, + "source": [ + "Aportación de estrato abierto de E_1 (6,2)" + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "id": "4079d302", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((1/3) * (s + 1)^-2, (t^6 + t^3 + 1)^-1)" + ] + }, + "execution_count": 30, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "F2t=F2(x=x+3)\n", + "Z2t = ZetaFunctions(F2t)\n", + "A2t=Z2t.topological_zeta(local=True, weights=[1,3,1])\n", + "M2t=Z2t.monodromy_zeta()\n", + "A2t, M2t" + ] + }, + { + "cell_type": "markdown", + "id": "95e60d58", + "metadata": {}, + "source": [ + "Tercera explosión E_3 (16,5)" + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "id": "69cffddc", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(x + 1,\n", + " x^4*y^10 + x^3*y^6*z + x^2*y^6*z + z^3,\n", + " y + 1,\n", + " x^10*y^6 + x^6*y^4*z + x^6*y^3*z + z^3)" + ] + }, + "execution_count": 31, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "g3=g2(x=x*y)/y\n", + "F3=F2(x=x*y)\n", + "g3b=g2(y=x*y)/x\n", + "F3b=F2(y=x*y)\n", + "g3, F3, g3b, F3b" + ] + }, + { + "cell_type": "code", + "execution_count": 32, + "id": "15da5854", + "metadata": {}, + "outputs": [ + { + "ename": "SyntaxError", + "evalue": "invalid syntax (1948788351.py, line 1)", + "output_type": "error", + "traceback": [ + "\u001b[0;36m Cell \u001b[0;32mIn [32], line 1\u001b[0;36m\u001b[0m\n\u001b[0;31m Aportación de intersección de E_1 (Integer(6),Integer(2)) y E_3 (Integer(16),Integer(5))\u001b[0m\n\u001b[0m ^\u001b[0m\n\u001b[0;31mSyntaxError\u001b[0m\u001b[0;31m:\u001b[0m invalid syntax\n" + ] + } + ], + "source": [ + "Aportación de intersección de E_1 (6,2) y E_3 (16,5)" + ] + }, + { + "cell_type": "code", + "execution_count": 33, + "id": "c2f28966", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((1/30) * (s + 1)^-2 * (s + 8/9)^-1 * (s^2 + 1/3*s + 8/3),\n", + " (-1) * (t - 1) * (t^2 + t + 1))" + ] + }, + "execution_count": 33, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Z13 = ZetaFunctions(F3)\n", + "A13=Z13.topological_zeta(local=True, weights=[2,5,1])\n", + "M13=Z13.monodromy_zeta()\n", + "A13, M13" + ] + }, + { + "cell_type": "markdown", + "id": "d5937892", + "metadata": {}, + "source": [ + "Aportación de intersección de E_2 (9,3) y E_3 (16,5)" + ] + }, + { + "cell_type": "code", + "execution_count": 34, + "id": "3b735845", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((1/30) * (s + 1)^-2 * (s + 8/9)^-1 * (s^2 + 1/3*s + 8/3),\n", + " (-1) * (t - 1) * (t^2 + t + 1))" + ] + }, + "execution_count": 34, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Z23 = ZetaFunctions(F3b)\n", + "A23=Z23.topological_zeta(local=True, weights=[5,3,1])\n", + "M23=Z23.monodromy_zeta()\n", + "A13, M23" + ] + }, + { + "cell_type": "markdown", + "id": "11af23cc", + "metadata": {}, + "source": [ + "Aportación de estrato abierto de E_3 (16,5)" + ] + }, + { + "cell_type": "code", + "execution_count": 35, + "id": "ac2bc045", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((-7/45) * (s + 8/9)^-1 * (s + 1)^-1 * (s - 8/7),\n", + " (-1) * (t - 1)^-1 * (t^2 + t + 1)^-1 * (t^6 + t^3 + 1)^-2)" + ] + }, + "execution_count": 35, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "F3t=F3(x=x+3)\n", + "Z3t = ZetaFunctions(F3t)\n", + "A3t=Z3t.topological_zeta(local=True, weights=[1,5,1])\n", + "M3t=Z3t.monodromy_zeta()\n", + "A3t, M3t" + ] + }, + { + "cell_type": "code", + "execution_count": 36, + "id": "634cf890", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((-7/45) * (s + 8/9)^-1 * (s + 1)^-1 * (s - 8/7),\n", + " (-1) * (t - 1)^-1 * (t^2 + t + 1)^-1 * (t^6 + t^3 + 1)^-2)" + ] + }, + "execution_count": 36, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "F3bt=F3b(y=y+3)\n", + "Z3bt = ZetaFunctions(F3bt)\n", + "A3bt=Z3bt.topological_zeta(local=True, weights=[5,1,1])\n", + "M3bt=Z3bt.monodromy_zeta()\n", + "A3bt, M3bt" + ] + }, + { + "cell_type": "markdown", + "id": "692b49ed", + "metadata": {}, + "source": [ + "Aportación de intersección de E_3 (16,5) y g_1" + ] + }, + { + "cell_type": "code", + "execution_count": 37, + "id": "bb1fe1fb", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((1/6) * (s + 8/9)^-1 * (s + 9/10)^-1 * (s + 1)^-1 * (s^2 + 142/75*s + 24/25),\n", + " (t + 1) * (t - 1)^2 * (t^2 + t + 1) * (t^4 - t^3 + t^2 - t + 1) * (t^4 + t^3 + t^2 + t + 1))" + ] + }, + "execution_count": 37, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "F30=F3(x=x-1)\n", + "Z30 = ZetaFunctions(F30)\n", + "A30=Z30.topological_zeta(local=True, weights=[1,5,1])\n", + "M30=Z30.monodromy_zeta()\n", + "A30, M30" + ] + }, + { + "cell_type": "code", + "execution_count": 38, + "id": "449e86f1", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((1/6) * (s + 8/9)^-1 * (s + 9/10)^-1 * (s + 1)^-1 * (s^2 + 142/75*s + 24/25),\n", + " (t + 1) * (t - 1)^2 * (t^2 + t + 1) * (t^4 - t^3 + t^2 - t + 1) * (t^4 + t^3 + t^2 + t + 1))" + ] + }, + "execution_count": 38, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "F3b0=F3b(y=y-1)\n", + "Z3b0 = ZetaFunctions(F3b0)\n", + "A3b0=Z3b0.topological_zeta(local=True, weights=[5,1,1])\n", + "M3b0=Z3b0.monodromy_zeta()\n", + "A3b0, M3b0" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "66af979c", + "metadata": {}, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "961e7c1f", + "metadata": {}, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": 49, + "id": "a269acd5", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0" + ] + }, + "execution_count": 49, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "T. = QQ[]\n", + "A=A1+A2b+A13+A23-A3t+A30\n", + "B=(-11/90) * (s + 8/9)^-1 * (s + 9/10)^-1 * (s + 1)^-1 * (s^2 - 69/11*s - 72/11)\n", + "A-B" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "7d561179", + "metadata": {}, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "5894ac92", + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "SageMath 10.2", + "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": "3.11.1" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} From b0a4438e27d69e97e9fd8f38638b682ffe9855ea Mon Sep 17 00:00:00 2001 From: Enrique Artal Date: Wed, 21 Feb 2024 20:02:03 +0100 Subject: [PATCH 3/6] estratos --- Estratos.ipynb | 1022 ++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 1022 insertions(+) create mode 100644 Estratos.ipynb diff --git a/Estratos.ipynb b/Estratos.ipynb new file mode 100644 index 0000000..c85b269 --- /dev/null +++ b/Estratos.ipynb @@ -0,0 +1,1022 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 18, + "id": "03414fb8-f6f5-4796-a227-6aeed863ab6c", + "metadata": {}, + "outputs": [], + "source": [ + "T. = QQ[]\n", + "S. = PolynomialRing(QQ, order='neglex')\n", + "R. = PolynomialRing(QQ, order='neglex')\n", + "f = u^2 * v^2 + u^5 + v^5\n", + "g = u^3 * v^3 + u^7 + v^7\n", + "F = z^3 - 3 * f(u=x, v=y) * z + 2 * g(u=x, v=y)" + ] + }, + { + "cell_type": "code", + "execution_count": 19, + "id": "c17bb619-1df9-4ea5-bfce-7f77cf56e00d", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[Ideal (z, y, x) of Multivariate Polynomial Ring in x, y, z over Rational Field]" + ] + }, + "execution_count": 19, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "F.jacobian_ideal().minimal_associated_primes()" + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "id": "f65646af-e328-430f-991d-10a0c2348fee", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "0" + ] + }, + "execution_count": 23, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "h = f^3 - g^2\n", + "F.discriminant(z)(x=u, y=v) - 108 * h" + ] + }, + { + "cell_type": "code", + "execution_count": 22, + "id": "00201766-5028-41ba-a13e-4740790a538e", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "65" + ] + }, + "execution_count": 22, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "F.jacobian_ideal().vector_space_dimension()" + ] + }, + { + "cell_type": "code", + "execution_count": 24, + "id": "d0ebfd8e-9c6f-4dc1-8230-9f3dfcaa2b4d", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "148" + ] + }, + "execution_count": 24, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "h.jacobian_ideal().vector_space_dimension()" + ] + }, + { + "cell_type": "code", + "execution_count": 27, + "id": "73aaa28f-0f0f-4e2e-8c8d-7c8473336e0e", + "metadata": {}, + "outputs": [], + "source": [ + "load('/home/artal/Descargas/ZetaFunctionsNewtonND.py')" + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "id": "0e83692e-907b-493a-8911-945c2925e586", + "metadata": {}, + "outputs": [], + "source": [ + "Z = ZetaFunctions(h)" + ] + }, + { + "cell_type": "code", + "execution_count": 47, + "id": "75c28fbb-c042-4afc-b79d-e60fbc149628", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "[(-1) * v^14,\n", + " (-1) * u^14,\n", + " (-2) * u^3 * v^10,\n", + " (-2) * v^3 * u^10,\n", + " (-1) * v^10 * (v^4 + 2*u^3),\n", + " (-1) * u^10 * (2*v^3 + u^4),\n", + " (-1) * (v + u) * u^3 * v^3 * (2*v^6 - 5*u*v^5 + 5*u^2*v^4 - 5*u^3*v^3 + 5*u^4*v^2 - 5*u^5*v + 2*u^6)]" + ] + }, + "execution_count": 47, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "[p.factor() for p in Z.get_polyfaces_dictionary(compact=True)]" + ] + }, + { + "cell_type": "code", + "execution_count": 46, + "id": "9e5736a1-ba87-4f95-8dc0-0ab5b7151bbd", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "False" + ] + }, + "execution_count": 46, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Z.is_newton_degenerated(local=True)" + ] + }, + { + "cell_type": "code", + "execution_count": 38, + "id": "35654137-b9d0-46d6-b6e3-5d6261bf8271", + "metadata": {}, + "outputs": [ + { + "data": { + "image/png": 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", + "text/plain": [ + "Graphics object consisting of 20 graphics primitives" + ] + }, + "execution_count": 38, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "Z.newton_plot()" + ] + }, + { + "cell_type": "code", + "execution_count": 49, + "id": "361b687b-43e1-4db8-be55-2f6a180f10f0", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + " Facet 0: -42 + 4*v + 3*u >= 0\n", + " Facet 1: -13 + v + u >= 0\n", + " Facet 3: -42 + 3*v + 4*u >= 0\n" + ] + } + ], + "source": [ + "Z.give_info_facets(compact=True)" + ] + }, + { + "cell_type": "code", + "execution_count": 43, + "id": "3de0f7de-3ed6-4084-b6de-2a0fe4d95579", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "-v^14 - 2*u^3*v^10 + 3*u^4*v^9 + 3*u^9*v^4 - 2*u^10*v^3 - u^14" + ] + }, + "execution_count": 43, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "h1 = sum(h.monomial_coefficient(u^i * v^j) * u^i * v^j for i, j in h.exponents() if (i, j) == (14, 0) or (i, j) == (0, 14) or i + j == 13)\n", + "h1" + ] + }, + { + "cell_type": "code", + "execution_count": 44, + "id": "6db00060-6963-4782-a837-db6dc6adadc9", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "148" + ] + }, + "execution_count": 44, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "h1.jacobian_ideal().vector_space_dimension()" + ] + }, + { + "cell_type": "code", + "execution_count": 50, + "id": "70044743-6a40-4234-a9fa-fa9a87064058", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "-v^14 + v^15 + 3*u^2*v^12 - 2*u^3*v^10 + 3*u^4*v^9 + 3*u^5*v^10 + 4*u^7*v^7 + 3*u^9*v^4 - 2*u^10*v^3 + 3*u^10*v^5 + 3*u^12*v^2 - u^14 + u^15" + ] + }, + "execution_count": 50, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "h" + ] + }, + { + "cell_type": "code", + "execution_count": 68, + "id": "2a533b35-964e-4615-ad76-1afd397797b2", + "metadata": {}, + "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": "code", + "execution_count": 147, + "id": "effb40ba-8e60-43a1-8d89-ed8cee03ff1d", + "metadata": {}, + "outputs": [], + "source": [ + "zetatop=0\n", + "zetamon=1" + ] + }, + { + "cell_type": "markdown", + "id": "df0f6a80-d6c0-4b16-8306-4f5c81545a31", + "metadata": {}, + "source": [ + "$\\check{E}_1$" + ] + }, + { + "cell_type": "code", + "execution_count": 148, + "id": "8b9a319a-2351-425a-8476-c083a125e393", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((t + 1)^14 * (t^2 - t + 1)^7 * (t^6 - t^5 + t^4 - t^3 + t^2 - t + 1)^7,\n", + " (s + 9/14) * (s + 1))" + ] + }, + "execution_count": 148, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "zetamon1, zetatop1 = estrato(z^3 + x^2 * z^2 + x^7, -7, [2, 1, 1])\n", + "zetamon *= zetamon1\n", + "zetatop += zetatop1\n", + "zetamon1.factor(), zetatop1.denominator().factor()" + ] + }, + { + "cell_type": "markdown", + "id": "922c65e0-f073-4c4c-a5ba-953a4f997c07", + "metadata": {}, + "source": [ + "${E}_1\\cap E_{4,i}$" + ] + }, + { + "cell_type": "code", + "execution_count": 149, + "id": "a5c32283-70d6-4903-bc3f-e248d46dbb48", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((t - 1)^2 * (t^2 + t + 1)^2, (s + 9/14) * (s + 2/3) * (s + 1))" + ] + }, + "execution_count": 149, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "zetamon14, zetatop14 = estrato(z^3 + x^2 * y^7 * z^2 + x^7 * y^21, 2, [2, 7, 1])\n", + "zetamon *= zetamon14\n", + "zetatop += zetatop14\n", + "zetamon14.factor(), zetatop14.denominator().factor()" + ] + }, + { + "cell_type": "markdown", + "id": "08c192da-a9d5-4bc6-b834-c2ff504d6b5c", + "metadata": {}, + "source": [ + "${E}_1\\cap\\Delta$" + ] + }, + { + "cell_type": "code", + "execution_count": 150, + "id": "20bba556-4db2-4c7a-ac3d-8641ece608d4", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((t + 1)^-7 * (t^2 - t + 1)^-7, (s + 9/14) * (s + 1))" + ] + }, + "execution_count": 150, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "zetamon1d, zetatop1d = estrato(z^3 + x^2 * z^2 + x^7 * y, 7, [2, 1, 1])\n", + "zetamon *= zetamon1d\n", + "zetatop += zetatop1d\n", + "zetamon1d.factor(), zetatop1d.denominator().factor()" + ] + }, + { + "cell_type": "code", + "execution_count": 156, + "id": "1ec7e8f6-859f-4145-a86f-7a911923a478", + "metadata": {}, + "outputs": [], + "source": [ + "zetamone1= zetamon1*zetamon14*zetamon1d\n", + "zetatope1= zetatop1 + zetatop14 + zetatop1d" + ] + }, + { + "cell_type": "code", + "execution_count": 157, + "id": "319eeaf1-bc5a-474d-95b3-78de7f894c5c", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((t - 1)^2 * (t + 1)^7 * (t^2 + t + 1)^2 * (t^6 - t^5 + t^4 - t^3 + t^2 - t + 1)^7,\n", + " (s + 9/14) * (s + 2/3) * (s + 1))" + ] + }, + "execution_count": 157, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "zetamone1.factor(), zetatope1.denominator().factor()" + ] + }, + { + "cell_type": "markdown", + "id": "ae8fc0d1-552a-414a-914a-421964737158", + "metadata": {}, + "source": [ + "$\\check{E}_{2,i}$" + ] + }, + { + "cell_type": "code", + "execution_count": 153, + "id": "34e87485-b397-435a-9d4c-978735f1e394", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((t - 1)^2 * (t^12 - t^11 + t^9 - t^8 + t^6 - t^4 + t^3 - t + 1)^-2,\n", + " (s + 16/21) * (s + 1))" + ] + }, + "execution_count": 153, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "zetamon2, zetatop2 = estrato(z^3 + x^5 * z + x^7, 2, [3, 1, 1])\n", + "zetamon *= zetamon2\n", + "zetatop += zetatop2\n", + "zetamon2.factor(), zetatop2.denominator().factor()" + ] + }, + { + "cell_type": "code", + "execution_count": 154, + "id": "32307a2d-cb48-4ce4-8130-ddcee91a5b7e", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "x^12 - x^11 + x^9 - x^8 + x^6 - x^4 + x^3 - x + 1" + ] + }, + "execution_count": 154, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "cyclotomic_polynomial(21)" + ] + }, + { + "cell_type": "markdown", + "id": "50a665ae-6aab-4a09-95a8-820582d36efa", + "metadata": {}, + "source": [ + "${E}_{2,i}\\cap E_{3,i}$" + ] + }, + { + "cell_type": "code", + "execution_count": 155, + "id": "bf417692-9e86-42cb-9bab-7efcea3174cf", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((t - 1)^2 * (t^2 + t + 1)^2, (s + 29/42) * (s + 16/21) * (s + 1))" + ] + }, + "execution_count": 155, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "zetamon23, zetatop23= estrato(z^3 + x^5 * y^10 * z + x^7 * y^14, 2, [3, 5, 1])\n", + "zetamon *= zetamon23\n", + "zetatop += zetatop23\n", + "zetamon23.factor(), zetatop23.denominator().factor()" + ] + }, + { + "cell_type": "code", + "execution_count": 158, + "id": "e31a134f-9d4d-4438-abcd-a0ae6881a84f", + "metadata": {}, + "outputs": [], + "source": [ + "zetamone2= zetamon2*zetamon23\n", + "zetatope2= zetatop2 + zetatop23" + ] + }, + { + "cell_type": "code", + "execution_count": 159, + "id": "22248c64-3b38-4779-abd4-fa532827a1ca", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((t - 1)^4 * (t^2 + t + 1)^2 * (t^12 - t^11 + t^9 - t^8 + t^6 - t^4 + t^3 - t + 1)^-2,\n", + " (s + 29/42) * (s + 1))" + ] + }, + "execution_count": 159, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "zetamone2.factor(), zetatope2.denominator().factor()" + ] + }, + { + "cell_type": "markdown", + "id": "bf5405fe-df5b-48c2-973d-cb79fa6e51c8", + "metadata": {}, + "source": [ + "$\\check{E}_{3,i}$" + ] + }, + { + "cell_type": "code", + "execution_count": 160, + "id": "82e29430-e591-41fd-805e-024e718b91a4", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((t - 1)^-2 * (t^2 - t + 1)^2 * (t^12 - t^11 + t^9 - t^8 + t^6 - t^4 + t^3 - t + 1)^2 * (t^12 + t^11 - t^9 - t^8 + t^6 - t^4 - t^3 + t + 1)^2,\n", + " (s + 29/42) * (s + 1))" + ] + }, + "execution_count": 160, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "zetamon3, zetatop3 = estrato(z^3 + x^10 * z + x^14, -2, [5, 1, 1])\n", + "zetamon *= zetamon3\n", + "zetatop += zetatop3\n", + "zetamon3.factor(), zetatop3.denominator().factor()" + ] + }, + { + "cell_type": "code", + "execution_count": 161, + "id": "513c0649-9724-43ed-9f5c-e78cba17b735", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "x^12 + x^11 - x^9 - x^8 + x^6 - x^4 - x^3 + x + 1" + ] + }, + "execution_count": 161, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "cyclotomic_polynomial(42)" + ] + }, + { + "cell_type": "markdown", + "id": "7f5da880-9e24-45e7-9ab1-b5d8eebaf76c", + "metadata": {}, + "source": [ + "${E}_{3,i}\\cap E_{4,i}$" + ] + }, + { + "cell_type": "code", + "execution_count": 162, + "id": "fa3bd566-3f90-40c8-b3df-fc27e02fc3e0", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((t - 1)^2 * (t^2 + t + 1)^2, (s + 2/3) * (s + 29/42) * (s + 1))" + ] + }, + "execution_count": 162, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "zetamon34, zetatop34 = estrato(z^3 + x^14 * y^10 * z + x^21 * y^14, 2, [7, 5, 1])\n", + "zetamon *= zetamon34\n", + "zetatop += zetatop34\n", + "zetamon34.factor(), zetatop34.denominator().factor()" + ] + }, + { + "cell_type": "markdown", + "id": "4016536c-7dc4-417d-bb0f-496b738517a4", + "metadata": {}, + "source": [ + "${E}_{3,i}\\cap f$" + ] + }, + { + "cell_type": "code", + "execution_count": 163, + "id": "16a5e4d8-afb2-43bc-b4bd-ba9d7824e77b", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((t - 1)^2 * (t^2 - t + 1)^-2 * (t^12 - t^11 + t^9 - t^8 + t^6 - t^4 + t^3 - t + 1)^-2 * (t^12 + t^11 - t^9 - t^8 + t^6 - t^4 - t^3 + t + 1)^-2,\n", + " (s + 29/42) * (s + 1))" + ] + }, + "execution_count": 163, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "zetamon3f, zetatop3f = estrato(z^3 + x * y^10 * z + y^14, 2, [1, 5, 1])\n", + "zetamon *= zetamon3f\n", + "zetatop += zetatop3f\n", + "zetamon3f.factor(), zetatop3f.denominator().factor()" + ] + }, + { + "cell_type": "code", + "execution_count": 164, + "id": "5468f2df-38ec-4823-9377-31ab2bc6dc11", + "metadata": {}, + "outputs": [], + "source": [ + "zetamone23= zetamone2 * zetamon3 * zetamon34 * zetamon3f\n", + "zetatope23= zetatope2 + zetatop3 + zetatop34 + zetatop3f" + ] + }, + { + "cell_type": "code", + "execution_count": 165, + "id": "13789e6d-cbba-4fac-93cf-10551da0979a", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((t - 1)^6 * (t^2 + t + 1)^4 * (t^12 - t^11 + t^9 - t^8 + t^6 - t^4 + t^3 - t + 1)^-2,\n", + " (s + 2/3) * (s + 1))" + ] + }, + "execution_count": 165, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "zetamone23.factor(), zetatope23.denominator().factor()" + ] + }, + { + "cell_type": "markdown", + "id": "ccc32acb-28ef-4628-a51e-3aff997646e9", + "metadata": {}, + "source": [ + "$\\check{E}_{4,i}$" + ] + }, + { + "cell_type": "code", + "execution_count": 166, + "id": "9efeade1-d21c-4ff9-ad19-0b67ddbbab82", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((t - 1)^4 * (t^2 + t + 1)^4 * (t^6 + t^5 + t^4 + t^3 + t^2 + t + 1)^8 * (t^12 - t^11 + t^9 - t^8 + t^6 - t^4 + t^3 - t + 1)^8,\n", + " (s + 2/3) * (s + 1))" + ] + }, + "execution_count": 166, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "zetamon4, zetatop4 = estrato(z^3 + x^14 * z + x^21, -4, [7, 1, 1])\n", + "zetamon *= zetamon4\n", + "zetatop += zetatop4\n", + "zetamon4.factor(), zetatop4.denominator().factor()" + ] + }, + { + "cell_type": "markdown", + "id": "a564ec00-2a96-47c8-97ff-5a5f500c626a", + "metadata": {}, + "source": [ + "${E}_{4,i}\\cap g$" + ] + }, + { + "cell_type": "code", + "execution_count": 167, + "id": "9dcde2d8-2a61-48a6-87e8-98ffb547ef16", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((t - 1)^-2 * (t^2 + t + 1)^-2 * (t^6 + t^5 + t^4 + t^3 + t^2 + t + 1)^-4 * (t^12 - t^11 + t^9 - t^8 + t^6 - t^4 + t^3 - t + 1)^-4,\n", + " (s + 2/3) * (s + 1))" + ] + }, + "execution_count": 167, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "zetamon4g, zetatop4g = estrato(z^3 + x^14 * z + x^21 * y, 2, [7, 1, 1])\n", + "zetamon *= zetamon4g\n", + "zetatop += zetatop4g\n", + "zetamon4g.factor(), zetatop4g.denominator().factor()" + ] + }, + { + "cell_type": "markdown", + "id": "2981a168-ed29-473e-bc38-4c6864ed7b26", + "metadata": {}, + "source": [ + "${E}_4\\cap\\Delta$" + ] + }, + { + "cell_type": "code", + "execution_count": 168, + "id": "767c1c15-0e88-492f-950a-82a54480a2eb", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((t^6 + t^5 + t^4 + t^3 + t^2 + t + 1)^-2 * (t^12 - t^11 + t^9 - t^8 + t^6 - t^4 + t^3 - t + 1)^-2,\n", + " (s + 2/3) * (s + 1))" + ] + }, + "execution_count": 168, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "zetamon4d, zetatop4d = estrato(z^3 + x^7 * z^2 + x^21 * y, 2, [7, 1, 1])\n", + "zetamon *= zetamon4d\n", + "zetatop += zetatop4d\n", + "zetamon4d.factor(), zetatop4d.denominator().factor()" + ] + }, + { + "cell_type": "code", + "execution_count": 172, + "id": "5c0f5123-59a7-4660-a298-d0aa238d5694", + "metadata": {}, + "outputs": [], + "source": [ + "zetamone4= zetamone23 * zetamon4 * zetamon4g * zetamon4d * zetamon14\n", + "zetatope4= zetatope23 + zetatop4 + zetatop4g + zetatop4d + zetatop14" + ] + }, + { + "cell_type": "code", + "execution_count": 173, + "id": "7323db86-9f16-4ded-a39e-e377836e617e", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((t - 1)^10 * (t^2 + t + 1)^8 * (t^6 + t^5 + t^4 + t^3 + t^2 + t + 1)^2,\n", + " (s + 9/14) * (s + 2/3) * (s + 1))" + ] + }, + "execution_count": 173, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "zetamone4.factor(), zetatope4.denominator().factor()" + ] + }, + { + "cell_type": "code", + "execution_count": 143, + "id": "b1d36f9c-e580-410d-a41e-90915236f8d7", + "metadata": {}, + "outputs": [], + "source": [ + "zetamon /= (t^3 - 1)^7" + ] + }, + { + "cell_type": "code", + "execution_count": 144, + "id": "3e2c75ac-c59e-48ce-8897-b25dd1283e6a", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "t^66 - t^63 + 5*t^59 - 5*t^56 + 8*t^52 - 8*t^49 - 14*t^38 + 14*t^35 - 14*t^31 + 14*t^28 + 8*t^17 - 8*t^14 + 5*t^10 - 5*t^7 + t^3 - 1" + ] + }, + "execution_count": 144, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "zetamon" + ] + }, + { + "cell_type": "code", + "execution_count": 145, + "id": "2698a314-3799-4832-a9dc-f735c14a1a4a", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(s + 9/14) * (s + 2/3) * (s + 1)" + ] + }, + "execution_count": 145, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "zetatop.denominator().factor()" + ] + }, + { + "cell_type": "code", + "execution_count": 146, + "id": "29a016fb-3316-49b5-9720-843637017b4f", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(t - 1)^3 * (t + 1)^7 * (t^2 + t + 1) * (t^6 + t^5 + t^4 + t^3 + t^2 + t + 1)^2 * (t^6 - t^5 + t^4 - t^3 + t^2 - t + 1)^7" + ] + }, + "execution_count": 146, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "zetamon.factor()" + ] + }, + { + "cell_type": "code", + "execution_count": 175, + "id": "274931c8-c536-4f97-9a5d-06a596a92285", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "1" + ] + }, + "execution_count": 175, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "zetamon4 * zetamon4g^2" + ] + }, + { + "cell_type": "code", + "execution_count": 176, + "id": "dcc73bc3-9e6e-45fe-99ce-6e8b17f6b159", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "1" + ] + }, + "execution_count": 176, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "zetamon3 * zetamon3f" + ] + }, + { + "cell_type": "code", + "execution_count": 180, + "id": "4afe75a2-f9f6-4690-a09b-570ba614d71e", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(10/21) * (s + 2/3)^-1 * (s + 1)^-1 * (s + 6/5)" + ] + }, + "execution_count": 180, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "zetatope23.factor()" + ] + }, + { + "cell_type": "code", + "execution_count": 181, + "id": "db53f3eb-871a-41d4-bb61-1cda3410c6dd", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(13/21) * (s + 9/14)^-1 * (s + 2/3)^-1 * (s + 1)^-1 * (s^2 + 43/26*s + 9/13)" + ] + }, + "execution_count": 181, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "zetatope4.factor()" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "824c0056-d8c4-40d1-8327-4b2a0498149d", + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "SageMath 10.3.beta8", + "language": "sage", + "name": "sagemath-devel" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.12.1" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} From 9a745ef74ca0cc9f0a872f834e904deaca4b3a60 Mon Sep 17 00:00:00 2001 From: "m.gonzalez" Date: Fri, 23 Feb 2024 17:31:53 +0100 Subject: [PATCH 4/6] archivo detalla primera explosion del ejemplo en archivo estratos --- detalleestratos.ipynb | 1184 +++++++++++++++++++++++++++++++++++++++++ 1 file changed, 1184 insertions(+) create mode 100644 detalleestratos.ipynb diff --git a/detalleestratos.ipynb b/detalleestratos.ipynb new file mode 100644 index 0000000..b50653e --- /dev/null +++ b/detalleestratos.ipynb @@ -0,0 +1,1184 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": 39, + "id": "b7e27f4d-8435-426c-a780-06e2e045718d", + "metadata": {}, + "outputs": [], + "source": [ + "reset()\n", + "load('ZetaFunctionsNewtonND.py')\n", + "R. = QQ[]\n", + "F=z^3 -3*(x^2*y^2 + x^5 + y^5)*z + 2*x^3*y^3 + x^7 + y^7\n", + "D=F.discriminant(z)\n", + "ZF = ZetaFunctions(F)" + ] + }, + { + "cell_type": "code", + "execution_count": 40, + "id": "c8dfedd2-f16f-4e50-acf7-e14b44374c6c", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "True" + ] + }, + "execution_count": 40, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "ZF.is_newton_degenerated()" + ] + }, + { + "cell_type": "code", + "execution_count": 41, + "id": "8786b09c-f4d6-440a-be82-30a26e6530f9", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "108*x^15 + 324*x^10*y^5 + 324*x^5*y^10 + 108*y^15 - 27*x^14 + 324*x^12*y^2 + 594*x^7*y^7 + 324*x^2*y^12 - 27*y^14 - 108*x^10*y^3 + 324*x^9*y^4 + 324*x^4*y^9 - 108*x^3*y^10" + ] + }, + "execution_count": 41, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "D" + ] + }, + { + "cell_type": "code", + "execution_count": 42, + "id": "a5ecaeee-9dc0-4491-8312-4ca22b364e2b", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Newton's polyhedron of 108*x^15 + 324*x^10*y^5 + 324*x^5*y^10 + 108*y^15 - 27*x^14 + 324*x^12*y^2 + 594*x^7*y^7 + 324*x^2*y^12 - 27*y^14 - 108*x^10*y^3 + 324*x^9*y^4 + 324*x^4*y^9 - 108*x^3*y^10:\n", + " support points = [(15, 0, 0), (10, 5, 0), (5, 10, 0), (0, 15, 0), (14, 0, 0), (12, 2, 0), (7, 7, 0), (2, 12, 0), (0, 14, 0), (10, 3, 0), (9, 4, 0), (4, 9, 0), (3, 10, 0)]\n", + " vertices = [(0, 14, 0), (14, 0, 0), (3, 10, 0), (10, 3, 0)]\n", + " number of proper faces = 19\n", + " Facet 0: 3*x + 4*y - 42 >= 0\n", + " Facet 1: x + y - 13 >= 0\n", + " Facet 2: x >= 0\n", + " Facet 3: 4*x + 3*y - 42 >= 0\n", + " Facet 4: y >= 0\n", + " Facet 5: z >= 0\n" + ] + } + ], + "source": [ + "d = ZetaFunctions(D)\n", + "d.give_info_newton()" + ] + }, + { + "cell_type": "code", + "execution_count": 43, + "id": "90017032-05aa-409e-bb72-75d8af9f5ff3", + "metadata": {}, + "outputs": [], + "source": [ + "F1=F(x=x*y)\n", + "D1=D(x=x*y)/y^13\n", + "F1b=F(y=x*y)\n", + "D1b=D(y=x*y)/x^13\n", + "ZF1 = ZetaFunctions(F1)\n", + "ZF1b = ZetaFunctions(F1b)" + ] + }, + { + "cell_type": "code", + "execution_count": 44, + "id": "d0bbe6c1-c702-44d1-add6-b2f4ce8097ed", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "((-108) * (x + 1) * (x^6 - 4*x^5 + 4*x^4 - 4*x^3 + 4*x^2 - 4*x + 1),\n", + " (-108) * (y + 1) * (y^6 - 4*y^5 + 4*y^4 - 4*y^3 + 4*y^2 - 4*y + 1))" + ] + }, + "execution_count": 44, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "d1=D1(y=0)\n", + "d1b=D1b(x=0)\n", + "(d1/x^3).factor(),(d1b/y^3).factor()" + ] + }, + { + "cell_type": "code", + "execution_count": 45, + "id": "6c34d700-f2ef-45f6-87c2-56d03b78f04c", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(True, True)" + ] + }, + "execution_count": 45, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "F1t=F1(x=x+1)\n", + "F1bt=F1b(y=y+1)\n", + "ZF1t = ZetaFunctions(F1t)\n", + "ZF1bt = ZetaFunctions(F1bt)\n", + "ZF1t.is_newton_degenerated(local=True), ZF1bt.is_newton_degenerated(local=True)" + ] + }, + { + "cell_type": "code", + "execution_count": 46, + "id": "53529479-e3ee-4d7f-ba62-afd9e7eeffb3", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Newton's polyhedron of x^7*y^7 + 7*x^6*y^7 + 21*x^5*y^7 + 35*x^4*y^7 - 3*x^5*y^5*z + 35*x^3*y^7 - 15*x^4*y^5*z + 2*x^3*y^6 + 21*x^2*y^7 - 30*x^3*y^5*z + 6*x^2*y^6 + 7*x*y^7 - 30*x^2*y^5*z + 6*x*y^6 + 2*y^7 - 3*x^2*y^4*z - 15*x*y^5*z + 2*y^6 - 6*x*y^4*z - 6*y^5*z - 3*y^4*z + z^3:\n", + " support points = [(7, 7, 0), (6, 7, 0), (5, 7, 0), (4, 7, 0), (5, 5, 1), (3, 7, 0), (4, 5, 1), (3, 6, 0), (2, 7, 0), (3, 5, 1), (2, 6, 0), (1, 7, 0), (2, 5, 1), (1, 6, 0), (0, 7, 0), (2, 4, 1), (1, 5, 1), (0, 6, 0), (1, 4, 1), (0, 5, 1), (0, 4, 1), (0, 0, 3)]\n", + " vertices = [(0, 0, 3), (0, 6, 0)]\n", + " number of proper faces = 11\n", + " Facet 0: x >= 0\n", + " Facet 1: y >= 0\n", + " Facet 2: y + 2*z - 6 >= 0\n", + " Facet 3: z >= 0\n" + ] + } + ], + "source": [ + "ZF1t.give_info_newton()" + ] + }, + { + "cell_type": "code", + "execution_count": 47, + "id": "3c518be7-15e0-462d-9546-68094910993b", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Newton's polyhedron of x^7*y^7 + 7*x^7*y^6 + 21*x^7*y^5 + 35*x^7*y^4 - 3*x^5*y^5*z + 35*x^7*y^3 - 15*x^5*y^4*z + 21*x^7*y^2 + 2*x^6*y^3 - 30*x^5*y^3*z + 7*x^7*y + 6*x^6*y^2 - 30*x^5*y^2*z + 2*x^7 + 6*x^6*y - 15*x^5*y*z - 3*x^4*y^2*z + 2*x^6 - 6*x^5*z - 6*x^4*y*z - 3*x^4*z + z^3:\n", + " support points = [(7, 7, 0), (7, 6, 0), (7, 5, 0), (7, 4, 0), (5, 5, 1), (7, 3, 0), (5, 4, 1), (7, 2, 0), (6, 3, 0), (5, 3, 1), (7, 1, 0), (6, 2, 0), (5, 2, 1), (7, 0, 0), (6, 1, 0), (5, 1, 1), (4, 2, 1), (6, 0, 0), (5, 0, 1), (4, 1, 1), (4, 0, 1), (0, 0, 3)]\n", + " vertices = [(0, 0, 3), (6, 0, 0)]\n", + " number of proper faces = 11\n", + " Facet 0: x + 2*z - 6 >= 0\n", + " Facet 1: x >= 0\n", + " Facet 2: y >= 0\n", + " Facet 3: z >= 0\n" + ] + } + ], + "source": [ + "ZF1bt.give_info_newton()" + ] + }, + { + "cell_type": "code", + "execution_count": 48, + "id": "72e1d393-5e3d-4453-a7f8-8b67cebc291a", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(2*y^2 + z) * (-y^2 + z)^2" + ] + }, + "execution_count": 48, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "cF1t = 2*y^6 - 3*y^4*z + z^3\n", + "cF1t.factor()" + ] + }, + { + "cell_type": "code", + "execution_count": 49, + "id": "4c62a268-6bc1-4470-aa1d-b8feb75f38b7", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "True" + ] + }, + "execution_count": 49, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "NF1t=F1t(z=z+y^2)\n", + "ZNF1t = ZetaFunctions(NF1t)\n", + "ZNF1t.is_newton_degenerated(local=True)" + ] + }, + { + "cell_type": "code", + "execution_count": 50, + "id": "83481949-7133-4292-ad20-516a6dd7db82", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Newton's polyhedron of x^7*y^7 + 7*x^6*y^7 + 18*x^5*y^7 + 20*x^4*y^7 - 3*x^5*y^5*z + 5*x^3*y^7 - 15*x^4*y^5*z + 2*x^3*y^6 - 9*x^2*y^7 - 30*x^3*y^5*z + 3*x^2*y^6 - 8*x*y^7 - 30*x^2*y^5*z - 4*y^7 - 3*x^2*y^4*z - 15*x*y^5*z - 6*x*y^4*z - 6*y^5*z + 3*y^2*z^2 + z^3:\n", + " support points = [(7, 7, 0), (6, 7, 0), (5, 7, 0), (4, 7, 0), (5, 5, 1), (3, 7, 0), (4, 5, 1), (3, 6, 0), (2, 7, 0), (3, 5, 1), (2, 6, 0), (1, 7, 0), (2, 5, 1), (0, 7, 0), (2, 4, 1), (1, 5, 1), (1, 4, 1), (0, 5, 1), (0, 2, 2), (0, 0, 3)]\n", + " vertices = [(0, 0, 3), (0, 2, 2), (2, 6, 0), (0, 7, 0)]\n", + " number of proper faces = 17\n", + " Facet 0: y >= 0\n", + " Facet 1: x >= 0\n", + " Facet 2: x + 2*y + 5*z - 14 >= 0\n", + " Facet 3: y + 2*z - 6 >= 0\n", + " Facet 4: z >= 0\n" + ] + } + ], + "source": [ + "ZNF1t.give_info_newton()" + ] + }, + { + "cell_type": "code", + "execution_count": 51, + "id": "fbe580cd-1e04-4020-ac10-ff94f1b5f4fe", + "metadata": {}, + "outputs": [ + { + "data": { + "text/html": [ + "\n", + "\n" + ], + "text/plain": [ + "Graphics3d Object" + ] + }, + "execution_count": 51, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "ZNF1t.newton_plot()" + ] + }, + { + "cell_type": "code", + "execution_count": 52, + "id": "a45f7abe-34ff-40ad-8138-fa0f4b93c479", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "(3) * y^2 * (-x*y^2 + z)^2" + ] + }, + "execution_count": 52, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "aux=3*x^2*y^6 - 6*x*y^4*z + 3*y^2*z^2\n", + "aux.factor()" + ] + }, + { + "cell_type": "code", + "execution_count": 53, + "id": "15a419b5-6ca0-48a2-b77e-11d039aa52cc", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "False" + ] + }, + "execution_count": 53, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "NNF1t=NF1t(z=z+x*y^2)\n", + "ZNNF1t = ZetaFunctions(NNF1t)\n", + "ZNNF1t.is_newton_degenerated(local=True)" + ] + }, + { + "cell_type": "code", + "execution_count": 54, + "id": "64631350-3ee6-40db-a55e-f1847f8be665", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Newton's polyhedron of x^7*y^7 + 4*x^6*y^7 + 3*x^5*y^7 - 10*x^4*y^7 - 3*x^5*y^5*z - 25*x^3*y^7 - 15*x^4*y^5*z - 24*x^2*y^7 - 30*x^3*y^5*z - 14*x*y^7 - 30*x^2*y^5*z - 4*y^7 - 15*x*y^5*z - 6*y^5*z + 3*x*y^2*z^2 + 3*y^2*z^2 + z^3:\n", + " support points = [(7, 7, 0), (6, 7, 0), (5, 7, 0), (4, 7, 0), (5, 5, 1), (3, 7, 0), (4, 5, 1), (2, 7, 0), (3, 5, 1), (1, 7, 0), (2, 5, 1), (0, 7, 0), (1, 5, 1), (0, 5, 1), (1, 2, 2), (0, 2, 2), (0, 0, 3)]\n", + " vertices = [(0, 0, 3), (0, 2, 2), (0, 7, 0)]\n", + " number of proper faces = 15\n", + " Facet 0: y + 2*z - 6 >= 0\n", + " Facet 1: y >= 0\n", + " Facet 2: x >= 0\n", + " Facet 3: z >= 0\n", + " Facet 4: 2*y + 5*z - 14 >= 0\n" + ] + } + ], + "source": [ + "ZNNF1t.give_info_newton()" + ] + }, + { + "cell_type": "markdown", + "id": "d018605b-b591-496e-b279-3f668a118e79", + "metadata": {}, + "source": [ + "Traslaccion de un golpe" + ] + }, + { + "cell_type": "code", + "execution_count": 57, + "id": "094a930d-b77d-416b-a4e5-0257f8dc23ce", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "False" + ] + }, + "execution_count": 57, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "GF1t=F1t(z=z+y^2+x*y^2)\n", + "ZGF1t = ZetaFunctions(GF1t)\n", + "ZGF1t.is_newton_degenerated(local=True)" + ] + }, + { + "cell_type": "code", + "execution_count": 58, + "id": "6c768f51-4886-4e20-83fa-a17ea1ab3e53", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Newton's polyhedron of x^7*y^7 + 4*x^6*y^7 + 3*x^5*y^7 - 10*x^4*y^7 - 3*x^5*y^5*z - 25*x^3*y^7 - 15*x^4*y^5*z - 24*x^2*y^7 - 30*x^3*y^5*z - 14*x*y^7 - 30*x^2*y^5*z - 4*y^7 - 15*x*y^5*z - 6*y^5*z + 3*x*y^2*z^2 + 3*y^2*z^2 + z^3:\n", + " support points = [(7, 7, 0), (6, 7, 0), (5, 7, 0), (4, 7, 0), (5, 5, 1), (3, 7, 0), (4, 5, 1), (2, 7, 0), (3, 5, 1), (1, 7, 0), (2, 5, 1), (0, 7, 0), (1, 5, 1), (0, 5, 1), (1, 2, 2), (0, 2, 2), (0, 0, 3)]\n", + " vertices = [(0, 0, 3), (0, 2, 2), (0, 7, 0)]\n", + " number of proper faces = 15\n", + " Facet 0: y + 2*z - 6 >= 0\n", + " Facet 1: y >= 0\n", + " Facet 2: x >= 0\n", + " Facet 3: z >= 0\n", + " Facet 4: 2*y + 5*z - 14 >= 0\n" + ] + } + ], + "source": [ + "ZGF1t.give_info_newton()" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "588205d1-e7c5-4a6b-87f6-9128b5a2a2eb", + "metadata": {}, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "8cd120e8-7440-4453-927e-6c8137d680c4", + "metadata": {}, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "69465e5c-28b4-4f1b-b6a7-4132f41b4d43", + "metadata": {}, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": 22, + "id": "a097d5ea-0585-44b7-a912-04f36be90d51", + "metadata": {}, + "outputs": [ + { + "data": { + "text/plain": [ + "True" + ] + }, + "execution_count": 22, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "F1=z^3 -3*(((y^2+2*x^3)^2+x^5*y)*(((x^2+y^3)^2+y^7*x)))*z + 2*((y^4*x^13)*(x^4*y^13)) + x^40 + y^40\n", + "D1=F1.discriminant(z)\n", + "ZD1 = ZetaFunctions(D1)\n", + "ZD1.is_newton_degenerated()" + ] + }, + { + "cell_type": "code", + "execution_count": 23, + "id": "612d96ca-666e-4dcb-9ba5-84c705ecdd8e", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Newton's polyhedron of -27*x^80 - 54*x^40*y^40 - 27*y^80 - 108*x^57*y^17 - 108*x^17*y^57 - 108*x^34*y^34 + 6912*x^21*y^21 + 5184*x^20*y^22 + 1296*x^19*y^23 + 108*x^18*y^24 + 20736*x^18*y^23 + 10368*x^17*y^24 + 1296*x^16*y^25 + 20736*x^20*y^20 + 15552*x^19*y^21 + 3888*x^18*y^22 + 324*x^17*y^23 + 25920*x^15*y^25 + 7776*x^14*y^26 + 324*x^13*y^27 + 41472*x^22*y^17 + 31104*x^21*y^18 + 7776*x^20*y^19 + 648*x^19*y^20 + 62208*x^17*y^22 + 31104*x^16*y^23 + 3888*x^15*y^24 + 17280*x^12*y^27 + 2592*x^11*y^28 + 20736*x^24*y^14 + 15552*x^23*y^15 + 3888*x^22*y^16 + 324*x^21*y^17 + 145152*x^19*y^19 + 77760*x^18*y^20 + 11664*x^17*y^21 + 324*x^16*y^22 + 77760*x^14*y^24 + 23328*x^13*y^25 + 972*x^12*y^26 + 6480*x^9*y^29 + 324*x^8*y^30 + 145152*x^21*y^16 + 93312*x^20*y^17 + 19440*x^19*y^18 + 1296*x^18*y^19 + 217728*x^16*y^21 + 77760*x^15*y^22 + 5832*x^14*y^23 + 51840*x^11*y^26 + 7776*x^10*y^27 + 1296*x^6*y^31 + 124416*x^23*y^13 + 93312*x^22*y^14 + 23328*x^21*y^15 + 1944*x^20*y^16 + 333504*x^18*y^18 + 152928*x^17*y^19 + 17820*x^16*y^20 + 108*x^15*y^21 + 181440*x^13*y^23 + 38880*x^12*y^24 + 972*x^11*y^25 + 19440*x^8*y^28 + 972*x^7*y^29 + 108*x^3*y^33 + 82944*x^25*y^10 + 62208*x^24*y^11 + 15552*x^23*y^12 + 1296*x^22*y^13 + 414720*x^20*y^15 + 217728*x^19*y^16 + 31104*x^18*y^17 + 648*x^17*y^18 + 383616*x^15*y^20 + 111456*x^14*y^21 + 5184*x^13*y^22 + 90720*x^10*y^25 + 9720*x^9*y^26 + 3888*x^5*y^30 + 20736*x^27*y^7 + 15552*x^26*y^8 + 3888*x^25*y^9 + 324*x^24*y^10 + 352512*x^22*y^12 + 202176*x^21*y^13 + 34992*x^20*y^14 + 1620*x^19*y^15 + 590976*x^17*y^17 + 202176*x^16*y^18 + 13608*x^15*y^19 + 252720*x^12*y^22 + 39852*x^11*y^23 + 324*x^10*y^24 + 27216*x^7*y^27 + 972*x^6*y^28 + 324*x^2*y^32 + 200448*x^24*y^9 + 134784*x^23*y^10 + 29808*x^22*y^11 + 2160*x^21*y^12 + 622080*x^19*y^14 + 248832*x^18*y^15 + 23328*x^17*y^16 + 466560*x^14*y^19 + 93312*x^13*y^20 + 1944*x^12*y^21 + 98928*x^9*y^24 + 6480*x^8*y^25 + 4536*x^4*y^29 + 103680*x^26*y^6 + 77760*x^25*y^7 + 19440*x^24*y^8 + 1620*x^23*y^9 + 492480*x^21*y^11 + 230688*x^20*y^12 + 26892*x^19*y^13 + 596160*x^16*y^16 + 147744*x^15*y^17 + 4860*x^14*y^18 + 220320*x^11*y^21 + 21384*x^10*y^22 + 22356*x^6*y^26 + 324*x^5*y^27 + 324*x*y^31 + 41472*x^28*y^3 + 31104*x^27*y^4 + 7776*x^26*y^5 + 648*x^25*y^6 + 311040*x^23*y^8 + 155520*x^22*y^9 + 19440*x^21*y^10 + 570240*x^18*y^13 + 163296*x^17*y^14 + 6480*x^16*y^15 + 336960*x^13*y^18 + 42768*x^12*y^19 + 62208*x^8*y^23 + 1944*x^7*y^24 + 2592*x^3*y^28 + 6912*x^30 + 5184*x^29*y + 1296*x^28*y^2 + 108*x^27*y^3 + 124416*x^25*y^5 + 62208*x^24*y^6 + 7776*x^23*y^7 + 388800*x^20*y^10 + 116640*x^19*y^11 + 4860*x^18*y^12 + 365040*x^15*y^15 + 52812*x^14*y^16 + 112752*x^10*y^20 + 4860*x^9*y^21 + 9720*x^5*y^25 + 108*y^30 + 20736*x^27*y^2 + 10368*x^26*y^3 + 1296*x^25*y^4 + 155520*x^22*y^7 + 46656*x^21*y^8 + 1944*x^20*y^9 + 259200*x^17*y^12 + 38880*x^16*y^13 + 133488*x^12*y^17 + 6480*x^11*y^18 + 20736*x^7*y^22 + 648*x^2*y^27 + 25920*x^24*y^4 + 7776*x^23*y^5 + 324*x^22*y^6 + 103680*x^19*y^9 + 15552*x^18*y^10 + 97200*x^14*y^14 + 4860*x^13*y^15 + 26244*x^9*y^19 + 1620*x^4*y^24 + 17280*x^21*y^6 + 2592*x^20*y^7 + 38880*x^16*y^11 + 1944*x^15*y^12 + 19440*x^11*y^16 + 2160*x^6*y^21 + 6480*x^18*y^8 + 324*x^17*y^9 + 7776*x^13*y^13 + 1620*x^8*y^18 + 1296*x^15*y^10 + 648*x^10*y^15 + 108*x^12*y^12:\n", + " support points = [(80, 0, 0), (40, 40, 0), (0, 80, 0), (57, 17, 0), (17, 57, 0), (34, 34, 0), (21, 21, 0), (20, 22, 0), (19, 23, 0), (18, 24, 0), (18, 23, 0), (17, 24, 0), (16, 25, 0), (20, 20, 0), (19, 21, 0), (18, 22, 0), (17, 23, 0), (15, 25, 0), (14, 26, 0), (13, 27, 0), (22, 17, 0), (21, 18, 0), (20, 19, 0), (19, 20, 0), (17, 22, 0), (16, 23, 0), (15, 24, 0), (12, 27, 0), (11, 28, 0), (24, 14, 0), (23, 15, 0), (22, 16, 0), (21, 17, 0), (19, 19, 0), (18, 20, 0), (17, 21, 0), (16, 22, 0), (14, 24, 0), (13, 25, 0), (12, 26, 0), (9, 29, 0), (8, 30, 0), (21, 16, 0), (20, 17, 0), (19, 18, 0), (18, 19, 0), (16, 21, 0), (15, 22, 0), (14, 23, 0), (11, 26, 0), (10, 27, 0), (6, 31, 0), (23, 13, 0), (22, 14, 0), (21, 15, 0), (20, 16, 0), (18, 18, 0), (17, 19, 0), (16, 20, 0), (15, 21, 0), (13, 23, 0), (12, 24, 0), (11, 25, 0), (8, 28, 0), (7, 29, 0), (3, 33, 0), (25, 10, 0), (24, 11, 0), (23, 12, 0), (22, 13, 0), (20, 15, 0), (19, 16, 0), (18, 17, 0), (17, 18, 0), (15, 20, 0), (14, 21, 0), (13, 22, 0), (10, 25, 0), (9, 26, 0), (5, 30, 0), (27, 7, 0), (26, 8, 0), (25, 9, 0), (24, 10, 0), (22, 12, 0), (21, 13, 0), (20, 14, 0), (19, 15, 0), (17, 17, 0), (16, 18, 0), (15, 19, 0), (12, 22, 0), (11, 23, 0), (10, 24, 0), (7, 27, 0), (6, 28, 0), (2, 32, 0), (24, 9, 0), (23, 10, 0), (22, 11, 0), (21, 12, 0), (19, 14, 0), (18, 15, 0), (17, 16, 0), (14, 19, 0), (13, 20, 0), (12, 21, 0), (9, 24, 0), (8, 25, 0), (4, 29, 0), (26, 6, 0), (25, 7, 0), (24, 8, 0), (23, 9, 0), (21, 11, 0), (20, 12, 0), (19, 13, 0), (16, 16, 0), (15, 17, 0), (14, 18, 0), (11, 21, 0), (10, 22, 0), (6, 26, 0), (5, 27, 0), (1, 31, 0), (28, 3, 0), (27, 4, 0), (26, 5, 0), (25, 6, 0), (23, 8, 0), (22, 9, 0), (21, 10, 0), (18, 13, 0), (17, 14, 0), (16, 15, 0), (13, 18, 0), (12, 19, 0), (8, 23, 0), (7, 24, 0), (3, 28, 0), (30, 0, 0), (29, 1, 0), (28, 2, 0), (27, 3, 0), (25, 5, 0), (24, 6, 0), (23, 7, 0), (20, 10, 0), (19, 11, 0), (18, 12, 0), (15, 15, 0), (14, 16, 0), (10, 20, 0), (9, 21, 0), (5, 25, 0), (0, 30, 0), (27, 2, 0), (26, 3, 0), (25, 4, 0), (22, 7, 0), (21, 8, 0), (20, 9, 0), (17, 12, 0), (16, 13, 0), (12, 17, 0), (11, 18, 0), (7, 22, 0), (2, 27, 0), (24, 4, 0), (23, 5, 0), (22, 6, 0), (19, 9, 0), (18, 10, 0), (14, 14, 0), (13, 15, 0), (9, 19, 0), (4, 24, 0), (21, 6, 0), (20, 7, 0), (16, 11, 0), (15, 12, 0), (11, 16, 0), (6, 21, 0), (18, 8, 0), (17, 9, 0), (13, 13, 0), (8, 18, 0), (15, 10, 0), (10, 15, 0), (12, 12, 0)]\n", + " vertices = [(0, 30, 0), (30, 0, 0), (12, 12, 0)]\n", + " number of proper faces = 15\n", + " Facet 0: 2*x + 3*y - 60 >= 0\n", + " Facet 1: 3*x + 2*y - 60 >= 0\n", + " Facet 2: x >= 0\n", + " Facet 3: y >= 0\n", + " Facet 4: z >= 0\n" + ] + } + ], + "source": [ + "ZD1.give_info_newton()" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "bbee0079-f404-4ad3-b8c6-b95dc34fdcf1", + "metadata": {}, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "beea9f68-34eb-4fbf-95c7-709e523b21b4", + "metadata": {}, + "outputs": [], + "source": [] + }, + { + "cell_type": "code", + "execution_count": 25, + "id": "b07929c2-0908-4573-8af4-c72ae8a175d1", + "metadata": {}, + "outputs": [ + { + "name": "stdout", + "output_type": "stream", + "text": [ + "Newton's polyhedron of -27*x^80 - 54*x^40*y^40 - 27*y^80 - 108*x^57*y^17 - 108*x^17*y^57 - 108*x^34*y^34 + 6912*x^21*y^21 + 5184*x^20*y^22 + 1296*x^19*y^23 + 108*x^18*y^24 + 20736*x^18*y^23 + 10368*x^17*y^24 + 1296*x^16*y^25 + 20736*x^20*y^20 + 15552*x^19*y^21 + 3888*x^18*y^22 + 324*x^17*y^23 + 25920*x^15*y^25 + 7776*x^14*y^26 + 324*x^13*y^27 + 41472*x^22*y^17 + 31104*x^21*y^18 + 7776*x^20*y^19 + 648*x^19*y^20 + 62208*x^17*y^22 + 31104*x^16*y^23 + 3888*x^15*y^24 + 17280*x^12*y^27 + 2592*x^11*y^28 + 20736*x^24*y^14 + 15552*x^23*y^15 + 3888*x^22*y^16 + 324*x^21*y^17 + 145152*x^19*y^19 + 77760*x^18*y^20 + 11664*x^17*y^21 + 324*x^16*y^22 + 77760*x^14*y^24 + 23328*x^13*y^25 + 972*x^12*y^26 + 6480*x^9*y^29 + 324*x^8*y^30 + 145152*x^21*y^16 + 93312*x^20*y^17 + 19440*x^19*y^18 + 1296*x^18*y^19 + 217728*x^16*y^21 + 77760*x^15*y^22 + 5832*x^14*y^23 + 51840*x^11*y^26 + 7776*x^10*y^27 + 1296*x^6*y^31 + 124416*x^23*y^13 + 93312*x^22*y^14 + 23328*x^21*y^15 + 1944*x^20*y^16 + 333504*x^18*y^18 + 152928*x^17*y^19 + 17820*x^16*y^20 + 108*x^15*y^21 + 181440*x^13*y^23 + 38880*x^12*y^24 + 972*x^11*y^25 + 19440*x^8*y^28 + 972*x^7*y^29 + 108*x^3*y^33 + 82944*x^25*y^10 + 62208*x^24*y^11 + 15552*x^23*y^12 + 1296*x^22*y^13 + 414720*x^20*y^15 + 217728*x^19*y^16 + 31104*x^18*y^17 + 648*x^17*y^18 + 383616*x^15*y^20 + 111456*x^14*y^21 + 5184*x^13*y^22 + 90720*x^10*y^25 + 9720*x^9*y^26 + 3888*x^5*y^30 + 20736*x^27*y^7 + 15552*x^26*y^8 + 3888*x^25*y^9 + 324*x^24*y^10 + 352512*x^22*y^12 + 202176*x^21*y^13 + 34992*x^20*y^14 + 1620*x^19*y^15 + 590976*x^17*y^17 + 202176*x^16*y^18 + 13608*x^15*y^19 + 252720*x^12*y^22 + 39852*x^11*y^23 + 324*x^10*y^24 + 27216*x^7*y^27 + 972*x^6*y^28 + 324*x^2*y^32 + 200448*x^24*y^9 + 134784*x^23*y^10 + 29808*x^22*y^11 + 2160*x^21*y^12 + 622080*x^19*y^14 + 248832*x^18*y^15 + 23328*x^17*y^16 + 466560*x^14*y^19 + 93312*x^13*y^20 + 1944*x^12*y^21 + 98928*x^9*y^24 + 6480*x^8*y^25 + 4536*x^4*y^29 + 103680*x^26*y^6 + 77760*x^25*y^7 + 19440*x^24*y^8 + 1620*x^23*y^9 + 492480*x^21*y^11 + 230688*x^20*y^12 + 26892*x^19*y^13 + 596160*x^16*y^16 + 147744*x^15*y^17 + 4860*x^14*y^18 + 220320*x^11*y^21 + 21384*x^10*y^22 + 22356*x^6*y^26 + 324*x^5*y^27 + 324*x*y^31 + 41472*x^28*y^3 + 31104*x^27*y^4 + 7776*x^26*y^5 + 648*x^25*y^6 + 311040*x^23*y^8 + 155520*x^22*y^9 + 19440*x^21*y^10 + 570240*x^18*y^13 + 163296*x^17*y^14 + 6480*x^16*y^15 + 336960*x^13*y^18 + 42768*x^12*y^19 + 62208*x^8*y^23 + 1944*x^7*y^24 + 2592*x^3*y^28 + 6912*x^30 + 5184*x^29*y + 1296*x^28*y^2 + 108*x^27*y^3 + 124416*x^25*y^5 + 62208*x^24*y^6 + 7776*x^23*y^7 + 388800*x^20*y^10 + 116640*x^19*y^11 + 4860*x^18*y^12 + 365040*x^15*y^15 + 52812*x^14*y^16 + 112752*x^10*y^20 + 4860*x^9*y^21 + 9720*x^5*y^25 + 108*y^30 + 20736*x^27*y^2 + 10368*x^26*y^3 + 1296*x^25*y^4 + 155520*x^22*y^7 + 46656*x^21*y^8 + 1944*x^20*y^9 + 259200*x^17*y^12 + 38880*x^16*y^13 + 133488*x^12*y^17 + 6480*x^11*y^18 + 20736*x^7*y^22 + 648*x^2*y^27 + 25920*x^24*y^4 + 7776*x^23*y^5 + 324*x^22*y^6 + 103680*x^19*y^9 + 15552*x^18*y^10 + 97200*x^14*y^14 + 4860*x^13*y^15 + 26244*x^9*y^19 + 1620*x^4*y^24 + 17280*x^21*y^6 + 2592*x^20*y^7 + 38880*x^16*y^11 + 1944*x^15*y^12 + 19440*x^11*y^16 + 2160*x^6*y^21 + 6480*x^18*y^8 + 324*x^17*y^9 + 7776*x^13*y^13 + 1620*x^8*y^18 + 1296*x^15*y^10 + 648*x^10*y^15 + 108*x^12*y^12:\n", + " support points = [(80, 0, 0), (40, 40, 0), (0, 80, 0), (57, 17, 0), (17, 57, 0), (34, 34, 0), (21, 21, 0), (20, 22, 0), (19, 23, 0), (18, 24, 0), (18, 23, 0), (17, 24, 0), (16, 25, 0), (20, 20, 0), (19, 21, 0), (18, 22, 0), (17, 23, 0), (15, 25, 0), (14, 26, 0), (13, 27, 0), (22, 17, 0), (21, 18, 0), (20, 19, 0), (19, 20, 0), (17, 22, 0), (16, 23, 0), (15, 24, 0), (12, 27, 0), (11, 28, 0), (24, 14, 0), (23, 15, 0), (22, 16, 0), (21, 17, 0), (19, 19, 0), (18, 20, 0), (17, 21, 0), (16, 22, 0), (14, 24, 0), (13, 25, 0), (12, 26, 0), (9, 29, 0), (8, 30, 0), (21, 16, 0), (20, 17, 0), (19, 18, 0), (18, 19, 0), (16, 21, 0), (15, 22, 0), (14, 23, 0), (11, 26, 0), (10, 27, 0), (6, 31, 0), (23, 13, 0), (22, 14, 0), (21, 15, 0), (20, 16, 0), (18, 18, 0), (17, 19, 0), (16, 20, 0), (15, 21, 0), (13, 23, 0), (12, 24, 0), (11, 25, 0), (8, 28, 0), (7, 29, 0), (3, 33, 0), (25, 10, 0), (24, 11, 0), (23, 12, 0), (22, 13, 0), (20, 15, 0), (19, 16, 0), (18, 17, 0), (17, 18, 0), (15, 20, 0), (14, 21, 0), (13, 22, 0), (10, 25, 0), (9, 26, 0), (5, 30, 0), (27, 7, 0), (26, 8, 0), (25, 9, 0), (24, 10, 0), (22, 12, 0), (21, 13, 0), (20, 14, 0), (19, 15, 0), (17, 17, 0), (16, 18, 0), (15, 19, 0), (12, 22, 0), (11, 23, 0), (10, 24, 0), (7, 27, 0), (6, 28, 0), (2, 32, 0), (24, 9, 0), (23, 10, 0), (22, 11, 0), (21, 12, 0), (19, 14, 0), (18, 15, 0), (17, 16, 0), (14, 19, 0), (13, 20, 0), (12, 21, 0), (9, 24, 0), (8, 25, 0), (4, 29, 0), (26, 6, 0), (25, 7, 0), (24, 8, 0), (23, 9, 0), (21, 11, 0), (20, 12, 0), (19, 13, 0), (16, 16, 0), (15, 17, 0), (14, 18, 0), (11, 21, 0), (10, 22, 0), (6, 26, 0), (5, 27, 0), (1, 31, 0), (28, 3, 0), (27, 4, 0), (26, 5, 0), (25, 6, 0), (23, 8, 0), (22, 9, 0), (21, 10, 0), (18, 13, 0), (17, 14, 0), (16, 15, 0), (13, 18, 0), (12, 19, 0), (8, 23, 0), (7, 24, 0), (3, 28, 0), (30, 0, 0), (29, 1, 0), (28, 2, 0), (27, 3, 0), (25, 5, 0), (24, 6, 0), (23, 7, 0), (20, 10, 0), (19, 11, 0), (18, 12, 0), (15, 15, 0), (14, 16, 0), (10, 20, 0), (9, 21, 0), (5, 25, 0), (0, 30, 0), (27, 2, 0), (26, 3, 0), (25, 4, 0), (22, 7, 0), (21, 8, 0), (20, 9, 0), (17, 12, 0), (16, 13, 0), (12, 17, 0), (11, 18, 0), (7, 22, 0), (2, 27, 0), (24, 4, 0), (23, 5, 0), (22, 6, 0), (19, 9, 0), (18, 10, 0), (14, 14, 0), (13, 15, 0), (9, 19, 0), (4, 24, 0), (21, 6, 0), (20, 7, 0), (16, 11, 0), (15, 12, 0), (11, 16, 0), (6, 21, 0), (18, 8, 0), (17, 9, 0), (13, 13, 0), (8, 18, 0), (15, 10, 0), (10, 15, 0), (12, 12, 0)]\n", + " vertices = [(0, 30, 0), (30, 0, 0), (12, 12, 0)]\n", + " number of proper faces = 15\n", + " Facet 0: 2*x + 3*y - 60 >= 0\n", + " Facet 1: 3*x + 2*y - 60 >= 0\n", + " Facet 2: x >= 0\n", + " Facet 3: y >= 0\n", + " Facet 4: z >= 0\n" + ] + } + ], + "source": [ + "F2=z^3 -3*x^4+x^5*z + 2*x^6 + y^13\n", + "D2=F2.discriminant(z)\n", + "ZD2 = ZetaFunctions(D1)\n", + "ZD2.is_newton_degenerated()\n", + "ZD2.give_info_newton()" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "id": "98bc922e-6e45-4fb4-890d-e5f3225966c9", + "metadata": {}, + "outputs": [], + "source": [] + } + ], + "metadata": { + "kernelspec": { + "display_name": "SageMath 10.2", + "language": "sage", + "name": "sagemath-st" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.9.18" + } + }, + "nbformat": 4, + "nbformat_minor": 5 +} From d9f4555843995cc7ba9f62e86f0e4e01939177de Mon Sep 17 00:00:00 2001 From: "m.gonzalez" Date: Wed, 28 Feb 2024 23:41:58 +0100 Subject: [PATCH 5/6] Subir archivos a '' MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit ejemplo en que la resolución de fgdisc es más larga que la de disc y la parte extra no aporta disc y la parte extra no aporta --- IMG_5493.JPG | Bin 0 -> 1628149 bytes 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