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.