@ -110,28 +110,86 @@ 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.
often be solved by reducing them to square systems, by respectively choosing a suitable square subsystem or adding equations. Morever, we will restrict ourselves to systems with
isolated solutions, i.e. zero-dimensional varieties.
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
$$\left\{(z_1,\ldots,z_n),~z_i=e^{\frac{2\pi i k}{d_i}}\text{ for }k=0,\ldots,d_i\text{ and }i=1,\ldots,n\right\} .$$
$F=(f_1,\ldots,f_n)$ 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 could build a start system of the same size and whose
polynomials have the same degrees, but whose solutions are easy to find, and thus can be used as starting points for the method.
For instance, the system $G=(g_1,\ldots g_n)$, where
$$ g_i(x_1,\ldots x_n)=x_i^{d_i}-1 ,$$
is such a system, since it has exactly the $D$ solutions
$$\left\{\left(e^{\frac{k_1}{d_1}2\pi i},\ldots,e^{\frac{k_n}{d_n}2\pi i}\right),\text{ for }0\leq k_i\leq d_i-1\,\text{ and }i=1,\ldots,n\right\} .$$
\subsection{Choosing the homotopy}
The deformation between the original system and the start system is a \textit{homotopy}, for instance one of the form
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.
The deformation between the original system and the start system is a \textit{homotopy}, for instance the convex combination of $F$ and $G$
where $x:=(x_1,\ldots,x_n)$ and $t\in[0,1].$ 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. Therefore, we can implicitly
define a curve $z(t)$ in $\C^n$ by the equation \begin{equation}\label{eq:h2} H(z(t),t)=0,\end{equation} so that in order to approximate the roots of $F$ it is enough to numerically track $z(t)$.
To do so, we derive the expression \eqref{eq:h2} with respect to $t$, and get the \textit{Davidenko Differential Equation}
This is a system of $n$ first-order differential equations, which can be solved numerically for $z(t)$ as an initial value problem, and is called \textit{path tracking}.
\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
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$z(t)$ for different roots
\begin{itemize}
\item never cross each other for $t>0$ (at $t=0$$F$ could have singular solutions), and
\itemhave no singularities, i.e. never cross each other for $t>0$ (at $t=0$,$F$ could have singular solutions), and
\item don't go to infinity for $t\to0$ ($F$ could have a solution at infinity),
\end{itemize}
we can employ the \textit{Gamma trick}:
we can employ the \textit{Gamma trick}: this consists in modifying the linear homotopy
\eqref{eq:h1} by susbtituting the parameter $t\in[0,1]$ with a complex curve $q(t)$ connecting $0$ and $1$:
$$ q(t)=\frac{\gamma t}{\gamma t+(1-t)} ,$$
where $\gamma\in(0,1)$ is a random complex parameter.This "probability one" procedure, i.e. for any particular system choosing $\gamma$ outside of a finite amount of lines through
the origin ensures that we get a good homotopy, basically because of the finiteness of the branch locus of the homotopy.
We now want to track down individual roots, following the solution paths from
a root $z_0$ of the start system by solving the initial value problem associated to the Davidenko differential equation \eqref{eq:dav} with starting value $z_0$ and
$t$ ranging from $1$ to $0$.
This will be done numerically, using a first-order predictor-corrector tracking method, which consists in first using Euler's method to get an approximation
$\widetilde{z}_i$, and then using Newton's method to correct it
using equation \eqref{eq:h2} so that it becomes a good approximation $z_i$ of the next value of the solution path.
\subsubsection{Predictor: Euler's method}
Recall that Euler's method consists in approximating the solution of the initial value problem associated to a first-order ordinary differential equations
% Braced system of equations below
\begin{equation*}
\left\{
\begin{aligned}
&\dot{z}=f(z,t)\\
&z(t_0)=z_0
\end{aligned}
\right.
\end{equation*}
by the sequence of points $(z_i)_{i\in\N}$ defined by the recurrence relation
$$ z_{i+1}=z_i+h\cdot f(z_i,t_i) ,$$
where $h$ is the step size.
In our case, we have $$f(z,t)=-\left(\frac{\partial H}{\partial z}(z,t)\right)^{-1}\frac{\partial H}{\partial t}(z,t)$$ and $t_0=1$, since we track from $1$ to $0$. For the same
reason, we set $$t_i=t_{i-1}-h.$$ We will also use a variable step size, based on the output of each iteration.
\subsubsection{Corrector: Newton's method}
Since we want to solve $$H(z,t)=0,$$ we can use Newton's method to improve the approximation $\widetilde{z_i}$ obtained by Euler's method to a solution of such equation.
This is done by moving towards the root of the tangent line of $H$ at the current approximation, or in other words through the iteration
