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 + \reversemarginpar{\color{blue} \textbf{#1}} + \framebox{\begin{minipage}[c]{0.95 \textwidth} + #2\end{minipage}}\vspace{5 mm}\par}} + +% Shortcuts for colors +\newcommand{\cb}{\color {blue}} +\newcommand{\crd}{\color {red}} + +% For segues +\definecolor{blue-violet}{rgb}{0.54, 0.17, 0.89} +\definecolor{Blue}{rgb}{0.01, 0.28, 1.0} +\definecolor{gGreen}{rgb}{0.2, 0.8, 0.2} +\definecolor{Green}{rgb}{0.04, 0.85, 0.32} + +\newcommand{\seg}[1]{\textcolor{blue-violet}{#1}} + +% For red comments on typos +\renewcommand{\!}[1]{{\color{red}\text{\Huge$\star$ }#1}} + +% For double slash +%\newcommand{\sslash}{\mathbin{/\mkern-6mu/}} + + +%%%%%%%%%%%%%%%%%%%%% + +\makeatletter +\def\@tocline#1#2#3#4#5#6#7{\relax + \ifnum #1>\c@tocdepth % then omit + \else + \par \addpenalty\@secpenalty\addvspace{#2}% + \begingroup \hyphenpenalty\@M + \@ifempty{#4}{% + \@tempdima\csname 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}} + + +\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 0000000..0fba1df Binary files /dev/null and b/NonisoMHS.pdf differ 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 +}