A Polynomial System from Differential Equations
Date: July 19th (Friday)
Time: 15:00-15:30
Abstract
For a class of cubic differential systems the necessary and
sufficient conditions for the origin to be a center were
derived by I. S. Kukles in 1944. These conditions were discovered
to be incomplete in the later 1980s with the aid of elimination
methods and computer algebra tools. Since then there have been
several attempts to establish the complete conditions without
success. The problem of completing the conditions can be reduced
partially to decomposing a large system of polynomial equations,
which can be done in principle by existing algorithms of
polynomial elimination. However, the occurring polynomials
are too large in terms of degree and number of terms to be
manageable. Various techniques and software tools have been
tried both automatically and interactively, but the complete
conditions have never been obtained.
In this talk, we explain the initial problem, present the
polynomial system generated by a program from Kukles'
differential systems, report some of our experiments, and
point out the major difficulties encountered in dealing with
the system. The generated polynomial system and some of its
subsystems will be made available electronically. We propose
the problem of solving these systems as an open challenge for
testing elimination algorithms and their implementations
and call publicly for solutions from algorithm and
software developers.
