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Cox_Ideals_notes.tex
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Cox_Ideals_notes.tex
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%
% Cox_Ideals_notes.tex
%
\documentclass[twoside]{amsart}
\usepackage{amssymb,latexsym}
\usepackage{MnSymbol}
\usepackage{times}
\usepackage{graphics}
\usepackage{tikz}
\usepackage{hyperref}
\hypersetup{colorlinks=true, urlcolor=blue}
%\usepackage{simpsons}
%\usepackage{epsdice}
\usepackage{staves}
\usetikzlibrary{matrix,arrows}
%\usepackage{graphics}
\oddsidemargin-0.85cm
\evensidemargin-0.65cm
\topmargin-2.05cm %I recommend adding these three lines to increase the
\textwidth19.05cm %amount of usable space on the page (and save trees)
\textheight25.05cm
\parindent0.0em
%This next line (when uncommented) allow you to use encapsulated
%postscript files for figures in your document
%\usepackage{epsfig}
%plain makes sure that we have page numbers
\pagestyle{plain}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{axiom}{Axiom}
\newtheorem{lemma}{Lemma}
\newtheorem{proposition}{Proposition}
\newtheorem{corollary}{Corollary}
\theoremstyle{definition}
\newtheorem{definition}{Definition}
\title{Notes and Solutions to \emph{Ideals, Varieties, and Algorithms} by David A. Cox, John Little, Donal O'Shea }
\author{
Ernest Yeung - M\"{u}nchen
}
\date{Marz 2014, jaro, primavera}
%This defines a new command \questionhead which takes one argument and
%prints out Question #. with some space.
\newcommand{\questionhead}[1]
{\bigskip\bigskip
\noindent{\small\bf Question #1.}
\bigskip}
\newcommand{\problemhead}[1]
{
\noindent{\small\bf Problem #1.}
}
\newcommand{\exercisehead}[1]
{ \smallskip
\noindent{\small\bf Exercise #1.}
}
\newcommand{\solutionhead}[1]
{
\noindent{\small\bf Solution #1.}
}
%-----------------------------------
\begin{document}
%-----------------------------------
\maketitle
From the beginning of 2016, I decided to cease all explicit crowdfunding for any of my materials on physics, math. I failed to raise \emph{any} funds from previous crowdfunding efforts. I decided that if I was going to live in \emph{abundance}, I must lose a scarcity attitude. I am committed to keeping all of my material \textbf{open-sourced}. I give all my stuff \emph{for free}.
In the beginning of 2017, I received a very generous donation from a reader from Norway who found these notes useful, through \emph{PayPal}. If you find these notes useful, feel free to donate directly and easily through \href{https://www.paypal.com/cgi-bin/webscr?cmd=_donations&business=ernestsaveschristmas%2bpaypal%40gmail%2ecom&lc=US&item_name=ernestyalumni¤cy_code=USD&bn=PP%2dDonationsBF%3abtn_donateCC_LG%2egif%3aNonHosted}{PayPal}, which won't go through a 3rd. party such as indiegogo, kickstarter, patreon. Otherwise, under the \emph{open-source MIT license}, feel free to copy, edit, paste, make your own versions, share, use as you wish.
\noindent gmail : ernestyalumni \\
linkedin : ernestyalumni \\
tumblr : ernestyalumni \\
twitter : ernestyalumni \\
youtube : ernestyalumni \\
David Cox, John Little, Donal O'Shea. \textbf{Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra}, Third Edition, Springer
\section{Geometry, Algebra, and Algorithms}
\subsection{Polynomials and Affine Space}
fields are important is that linear algebra works over \emph{any} field
\begin{definition}[2] set of all polynomials in $x_1 , \dots , x_n$ with coefficients in $k$, denoted $k[x_1, \dots , x_n]$
\end{definition}
polynomial $f$ \emph{divides} polynomial $g$ provided $g= fh$ for some $h \in k[x_1, \dots , x_n ]$
$k[x_1, \dots, x_n]$ satisfies all field axioms except for existence of multiplicative inverses; commutative ring, $k[x_1, \dots , x_n]$ \emph{polynomial ring}
\subsubsection*{Exercises for 1 }
\exercisehead{1}
$\mathbb{F}_2$ commutative ring since it's an abelian group under addition, commutative in multiplication, and multiplicative identity exists, namely $1$. It is a field since for $1\neq 0$, the multiplicative identity is $1$.
\exercisehead{2}
\begin{enumerate}
\item[(a)]
\item[(b)]
\item[(c)]
\end{enumerate}
\subsection{Affine Varieties }
\subsection{Parametrizations of Affine Varieties}
\subsection{Ideals}
\subsection{Polynomials of One Variable}
\section{Groebner Bases}
\subsection{Introduction}
\subsection{Orderings on the Monomials in $k[x_1, \dots , x_n]$ }
\subsection{A Division Algorithm in $k[x_1, \dots , x_n ]$ }
\subsection{Monomial Ideals and Dickson's Lemma }
\subsection{The Hilbert Basis Theorem and Groebner Bases}
\subsection{Properties of Groebner Bases}
\subsection{Buchberger's Algorithm}
\section{Elimination Theory}
\subsection{The Elimination and Extension Theorems}
\subsection{The Geometry of Elimination}
\section{The Algebra-Geometry Dictionary}
\subsection{Hilbert's Nullstellensatz}
\subsection{Radical Ideals and the Ideal-Variety Correspondence}
\section{Polynomial and Rational Functions on a Variety}
\subsection{Polynomial Mappings }
\section{Robotics and Automatic Geometric Theorem Proving}
\subsection{Geometric Description of Robots }
\end{document}