3:30 p.m., Friday
Andrew J. Sommese
University of Notre Dame
Numerical Primary Decomposition
My talk will start with a discussion and examples of the
highly structured sparse polynomial systems that arise in
engineering. I will discuss the robust numerical methods, based
on algebraic geometry and homotopy continuation, for finding
a finite set of solutions of a given polynomial system, which
includes all of its isolated solutions.
I will then turn to very recent joint work of myself, Jan
Verschelde, and Charles Wampler, which motivated by engineering
problems, develops numerical alorithms to study positive dimensional
solution components of polynomial systems.
These algorithms, based on algebraic geometry and the methods
discussed earlier in the talk, compute much of the geometric
information contained in the primary decomposition of the solution
set of a given polynomial system. In particular, the algorithms
lay out the decomposition of the solution set into irreducible components,
and give upper bounds for the multiplicities of the irreducible components,
with the upper bound of an irreducible component equal to one if
and only if the irreducible component is reduced.
The basic data in a numerical primary decomposition are generic points, that certify the existence of irreducible
components of the solution set their dimensions, and their degrees.
The algorithms, which make essential use of generic projection and
interpolation, can also be used to produce, for each irreducible
component of the reduced solution set, a finite number of polynomials
that vanish precisely on the irreducible component.