Polynomial systems

A Gröbner basis for an ideal under an elimination order reflects much of the
geometric structure of the variety defined by the ideal. We discuss how this
relationship can be used in decomposing polynomial systems (with or without
parameters) and in primary decomposition of ideals. As an application, we show
how this technique can be used in designing deterministic algorithms for
factoring univariate polynomials over finite fields, which aims to reduce the
factoring problem to a combinatorial one. The talk is based on joint work

The polyhedral homotopy method is known to be a powerful numerical method for
approximating all isolated solutions of a system of polynomial equations. We discuss a parallel implementation of the polyhedral homotopy method, a dynamic enumeration of all fine mixed cells which is used in constructing a family of polyhedral homotopy functions and extensions of the Hornor Scheme to multivariate
polynomials for efficient evaluation of a system of polynomials and their partial
derivatives in the polyhedral homotopy method.

In Numerical Algebraic Geometry, solution components of polynomial systems are characterized by witness points. Such nice points are computed efficiently by continuation methods.

In this talk, which is joint work with Wenyuan Wu and Jan Verschelde, I will outline progress on extending these methods to Partial Differential Equations.

I will describe the jet geometric picture of PDE in their jet space, to which

The solution set of a polynomial system decomposes into a union of
irreducible components. The set of polynomials imposes on each component a
positive integer known as the multiplicity of the component. This number is of
interest not only because of its meaning in applications but also because a
number of numerical methods have difficulty in problems where the multiplicity
of a component is greater than one. In this talk, I will discuss a numerical
algorithm for determining the multiplicity of a component of an algebraic set.