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\begin{center}
{\Large  The Center Variety of Polynomial Differential Systems} \\
\medskip
Abdul Salam Jarrah \\
New Mexico State University\\
\vspace{1.0cm}
{\Large Abstract}
\end{center}

\bigskip
Consider the two-dimensional differential system of the form
\begin{equation}\label{gs}
\frac{dx}{dt} = P(x,y), \;\;\;\;
\frac{dy}{dt} = Q(x,y) 
\end{equation} 
where $P$ and $Q$ are polynomials with complex coefficients and $x$ 
and $y$ are complex functions. Suppose $(0,0)$ is a singular point
of System (\ref{gs}). Poincar\'e
defined the notion of a center for real vector fields in 
the plane. He was interested in the problem of the stability
of the solar system. Dulac defined the notion of a center for
complex vector fields and gave necessary and sufficient
conditions for System (\ref{gs}) to have a center at the origin, 
for the case $\deg(P) = \deg(Q) =2$. 
Identifying those systems with a center 
at the origin is the so-called {\it center problem}. This problem
is still open for higher dimensional systems.

To solve the center problem for a given system (\ref{gs}),
one needs to compute the so-called Lyapunov quantities, which are
polynomials in the coefficients of $P$ and $Q$. In this talk,
we present an algorithm, using methods from computational algebra,
to find all those systems with a center at the origin, such that the linear
parts of $P$ and $Q$ form a nonsingular matrix over the complex numbers.
If we restrict ourselves to real systems, our algorithm finds all the 
systems with a center at the origin and an axis of symmetry through the origin. 


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