# Polynomial systems

Thursday, September 21, 2006 - 10:50am - 11:40am

Shuhong Gao (Clemson University)

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

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

Wednesday, October 25, 2006 - 1:40pm - 2:30pm

Masakazu Kojima (Tokyo Institute of Technology)

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.

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.

Friday, October 27, 2006 - 10:50am - 11:40am

Gregory Reid (University of Western Ontario)

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

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

Wednesday, September 20, 2006 - 3:00pm - 3:50pm

Daniel Bates (University of Minnesota, Twin Cities)

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.

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.