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\textbf{Approximating Roots to Polynomials} \\
\textbf{of Several Complex Variables}
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REU \\
Pennsylvania State University
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Chris Staskewicz
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\textbf{The Problem:}

Given $n$ polynomials in $n$ complex variables
with complex coefficients, we aim to solve the system
\begin{eqnarray*}
G_1(z_1, \ldots, z_n) & = & 0 \\
& \vdots & \\
G_n(z_1, \ldots, z_n) & = & 0
\end{eqnarray*}
using a \textbf{``homotopy''} from a similar system
\begin{eqnarray*}
K_1(z_1, \ldots, z_n) & = & 0 \\
& \vdots & \\
K_n(z_1, \ldots, z_n) & = & 0
\end{eqnarray*}
where the solution is known.

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\vspace{1cm}
\textbf{Homotopy Construction:}

We create a time parametrized family of polynomials,
$\{ P_i \}$, given by
\begin{eqnarray*}
P_1(z_1, \ldots, z_n; t) & = & (1-t)\,K_1 + t\,G_1 \\
& \vdots & \\
P_n(z_1, \ldots, z_n; t) & = & (1-t)\,K_n + t\,G_n
\end{eqnarray*}
where $0 \leq t \leq 1$.

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\vspace{1cm}
\textbf{Finding the Known System;} \\
\textbf{Newton Polyhedra:}

Consider one of the given polynomials, $G_i(z_1, \ldots, z_n)$.
Take the individual terms,
$a_{i,j}(z_1)^{i_1} \cdots (z_n)^{i_n}$,
and create the lattice points $(i_1, \ldots, i_n)$ which depend
upon the exponents of the variables.
The \underline{convex hull} of these points define a Newton
polyhedron in dimension $n$.

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\vspace{1cm}
For example, the Newton polyhedra for the system,
\begin{eqnarray*}
G_1(z_1, z_2) & = & (z_1)^2 + (z_2)^2 - 1 \\
G_2(z_1, z_2) & = & (z_1) \, (z_2) - 1
\end{eqnarray*}
is shown below.
\begin{center}
\epsfig{file=polyhedra1.eps}
\epsfig{file=polyhedra2.eps}
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\textbf{A Nice Result for $n=2$:}

From the given system, $\{ G_1, G_2 \}$, construct the
Newton polyhedra, $\{ C_1, C_2 \}$.  Then,
\[ \#~\textrm{of roots}~\leq
	A(C_1 + C_2) - A(C_1) - A(C_2) = m \]
where $A$ is the area of the polyhedra, and
$C_1 + C_2$ is the Minkowski sum.

The known system becomes,
\begin{eqnarray*}
K_1(z_1, z_2) & = & (z_1)^m - 1\\
K_2(z_1, z_2) & = & (z_2)^m - 1
\end{eqnarray*}
which has the \textbf{``roots of unity''} as the initial
roots,
\[ e^{\frac{2 k \pi i}{m}} \]
with $0 \leq k \leq m-1$.

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\textbf{Evaluation Procedure:}

\begin{enumerate}
\item Begin at $t=0$ and store the initial roots from
the known system in some set, $S$.

\item Increment time by $\Delta t$ and apply
Newton's method to the family, $\{ P_i \}$,
using the previous values from $S$.

\item Update the current values in $S$ using the values
from step $2$.

\item Repeat steps $2$ and $3$ unitl $t=1$.
\end{enumerate}

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\textbf{Complications:}

\begin{enumerate}
\item \underline{Root collisions}: if at any time during
the evaluation procedure, two or more roots in the set $S$
get \textbf{``too close''} to each other, Newton's method
$\rightarrow \infty$.
\item \underline{Too many initial roots}: depending upon
which system one begins with, there may be too many initial
roots at $t=0$.
\end{enumerate}

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\textbf{Resolution (Collisions):}

Under the homotopy map, colliding roots will converge to each
other, rotate by $90^{\circ}$, and then diverge from each
other.

\begin{enumerate}
\item Create a \textbf{``warning distance''} parameter which
halts the evaluation.

\item Adjust the evaluation procedure and predict the moment
of collision, $t_c$.
  \begin{itemize}
  \item Back up one time step $(t \rightarrow t - \Delta t)$,
  and compute a few more Newton steps with time fixed.

  \item Find the squared distance between the colliding
  roots, $d_1$.

  \item Incriment time by $t \rightarrow t + \frac{\Delta t}{4}$,
  compute a few more Newton steps with time fixed,
  and find another distance between the roots, $d_2$.
  \end{itemize}
\begin{center}
\epsfig{file=collide.eps}
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  \begin{itemize}
  \item By similar triangles,
  \[ t_c = \frac{( t+ \Delta t/4 )d_2}{d_1 - d_2} \]

  \item Multiply the colliding roots by $\pm i$ and replace
  the new roots into $S$.
  \end{itemize}
\item Continue the loop with $t \rightarrow t_c + \frac{\Delta t}{4}$
until another collision is detected.
\end{enumerate}

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\textbf{Resolution (Too Many Initial Roots):}

Most often, if one starts with the
\textbf{``roots of unity''}, there will be more initial
roots than actual roots.
This is due to the inequality in the Minkowski formula.
However, these \textbf{``fake''} roots \textbf{have yet}
to cause problems in the evaluation, as the distances
between them and the actual roots becomes large.

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\textbf{Example $\#1$ (Illustration of Root Collision):}

Consider the given polynomial in one real variable,
\[ G(x) = 13 - 6x + x^2 \]
We use,
\[ K(x) = 4 - 5x + x^2 \]
as the known polynomial with roots $\{ 1, 4 \}$.
The solution to $G(x) = 0$ is $x = 3 \pm 2i$.

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\epsfig{file=graph1var1.ps}

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\vspace{1cm}
\textbf{Example $\#2$}

Consider the given system,
\begin{eqnarray*}
G_1(z_1, z_2) & = & (z_1)^2 + (z_2)^2 - 1 \\
G_2(z_1, z_2) & = & (z_1) \, (z_2) - 1
\end{eqnarray*}
The maximum number of roots is,
\begin{eqnarray*}
\#~\textrm{of roots} & \leq & A(C_1 + C_2) - A(C_1) - A(C_2) \\
		     & =    & 6 - 2 - 0
\end{eqnarray*}
The known system becomes,
\begin{eqnarray*}
K_1(z_1, z_2) & = & (z_1)^4 - 1\\
K_2(z_1, z_2) & = & (z_2)^4 - 1
\end{eqnarray*}
Initial roots that converged to the actual roots:
\[ \left(
   \begin{array}{c}
	1 \\ 1
   \end{array}
   \right) ~~
   \left(
   \begin{array}{c}
	1 \\ -1
   \end{array}
   \right) ~~
   \left(
   \begin{array}{c}
	-1 \\ 1
   \end{array}
   \right) ~~
   \left(
   \begin{array}{c}
	-1 \\ -1
   \end{array}
   \right) \]

The actual roots to the given system:
\begin{eqnarray*}
   \left(
   \begin{array}{c}
	0.866+0.5i \\ 0.866-0.5i
   \end{array}
   \right) &
   \left(
   \begin{array}{c}
	-0.866+0.5i \\ -0.866-0.5i
   \end{array}
   \right) \\
& \\
   \left(
   \begin{array}{c}
	0.866-0.5i \\ 0.866+0.5i
   \end{array}
   \right) &
   \left(
   \begin{array}{c}
	-0.866-0.5i \\ -0.866+0.5i
   \end{array}
   \right)
\end{eqnarray*}

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