\documentclass[a4paper]{article}

\usepackage[T1]{fontenc}
\usepackage{textcomp}
\usepackage[english]{babel}
\usepackage{hyperref}
\usepackage{amsmath, amssymb, amsthm}
\usepackage{geometry}
\usepackage{tikz-cd}

% for including julia code
\usepackage{jlcode}

% Remove indentation globally
\setlength{\parindent}{0pt}
% Have blank lines between paragraphs
\usepackage[parfill]{parskip}

\hypersetup{
    colorlinks = true, % links instead of boxes
    urlcolor   = cyan, % external hyperlinks
    linkcolor  = blue, % internal links
    citecolor  = cyan   % citations
}

\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\A}{\mathbb{A}}
\newcommand{\Z}{\mathbb{Z}}

% use bullets for items
\renewcommand{\labelitemii}{$\circ$}
\renewcommand{\Im}{\operatorname{Im}}

\newcommand\numberthis{\addtocounter{equation}{1}\tag{\theequation}}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]

\theoremstyle{definition}
\newtheorem{definition}{Definition}[section]

\theoremstyle{definition}
\newtheorem{example}{Example}[section]

\theoremstyle{remark}
\newtheorem*{remark}{Remark}

\theoremstyle{definition}
\newtheorem{exercise}{Esercizio}[section]
\newtheorem*{exercise*}{Esercizio}

\begin{document}
\begin{titlepage}
    \begin{sffamily}
        \begin{large}
            \begin{center}
                \vbox to 100pt{%
                    \includegraphics[width=3cm]{cherubino}%
                \vfil}
            \end{center}
            \begin{center}
                \begin{Large}
                    \uppercase{Universit\`a degli studi di Pisa}
                \end{Large}\\
                \rule{9cm}{.4pt}\\
                \smallskip
                Dipartimento di Matematica\\
                \medskip
                Corso di Laurea Triennale in
                Matematica\\
                \bigskip\vfill
                \begin{Large}
                    Laboratorio Computazionale
                \end{Large}\\
                \bigskip\bigskip\vfil
                \begin{Huge}
                    Parallel Homotopy Continuation in Julia
                \end{Huge}
                \bigskip\vfill
                \begin{tabular}{ll}
                    \textbf{Studente:} & Francesco Minnocci\\
                    \textbf{Matricola:} & 600455
                \end{tabular}
            \end{center}
            \begin{center}
                \vfill
                \rule{9cm}{.4pt}\\
                \medskip
                \uppercase{Anno Accademico 2022 - 2023}\\
            \end{center}
        \end{large}
    \end{sffamily}
\end{titlepage}

\tableofcontents
\newpage

\section{Introduction}
Homotopy Continuation is a numerical method for solving systems of polynomial equations.
It is based on the idea of ”deforming” a given system of equations into a simpler one, whose
solutions are known, and then tracking the solutions of the original system as the deformation
is undone.

In this project, the method will be implemented in the Julia programming language, making use
of parallel computing in order to speed multiple root finding. The method is described in detail
in \cite{BertiniBook}, which was the primary source for this report.

\section{Homotopy Continuation}
We will only consider \textit{square} systems of polynomial equations, i.e. systems of $n$ polynomial equations in $n$ variables, although or over- or under-determined systems can
often be solved by reducing them to square systems, by respectively choosing a suitable square subsystem or adding equations.

There are many ways to choose the "simpler" system, from now on called a \textit{start system}, but in general we can observe that, by Bezout's theorem, a system
$F=(f_1,\ldots,f_n)$ has at most $D:=d_1\ldots d_n$ solutions, where $d_i$ is the degre of $f_i(x_1,\ldots,x_n)$. So, we can use as a start system $G=(g_1,\ldots g_n)$, where
$$ g_i(x_1,\ldots x_n)=x_i^{d_i}-1 .$$
Indeed, this system has exactly $D$ solutions
$$ \left\{(z_1,\ldots,z_n),~z_i=e^{\frac{2\pi i k}{d_i}}\text{ for }k=0,\ldots,d_i\text{ and }i=1,\ldots,n\right\} .$$
\subsection{Choosing the homotopy}
The deformation between the original system and the start system is a \textit{homotopy}, for instance one of the form
\begin{equation}\label{eq:h1} H(x;t)=(1-t)F(x)+tG(x) ,\end{equation}
where $x:=(x_1,\ldots,x_n).$ This is such that the roots of $H(x;0)=G(x)$ are known, and the roots of $H(x;1)=F(x)$ are the solutions of the original system.
\subsubsection{Gamma trick}
While \eqref{eq:h1} is a fine choice of a homotopy, it's not what it's called a \textit{good homotopy}: in order to ensure that the solution paths
\begin{itemize}
    \item never cross each other for $t>0$ (at $t=0$ $F$ could have singular solutions), and
    \item don't go to infinity for $t\to 0$ ($F$ could have a solution at infinity),
\end{itemize}
we can employ the \textit{Gamma trick}:
\subsection{Tracking down the roots}
\subsubsection{Davidenko differential equation}
\subsubsection{Predictor: Euler's method}
\subsubsection{Corrector: Newton's method}

\section{Parallelization}

\section{Implementation}
\subsection{Julia code}
\jlinputlisting[caption={solve.jl}]{../solve.jl}
\jlinputlisting[caption={start-system.jl}]{../start-system.jl}
\jlinputlisting[caption={homotopy.jl}]{../homotopy.jl}
\jlinputlisting[caption={homogenize.jl}]{../homogenize.jl}
\jlinputlisting[caption={euler-newton.jl}]{../euler-newton.jl}
\jlinputlisting[caption={adapt-step.jl}]{../adapt-step.jl}
\jlinputlisting[caption={plot.jl}]{../plot.jl}
\subsection{Hardware}

\section{Results}

\thebibliography{2}
\bibitem{BertiniBook} Bates, Daniel J. \textit{Numerically solving polynomial systems with Bertini}. SIAM, Society for Industrial Applied Mathematics, 2013.
\end{document}