Symmetric Functions Applied to Decomposing Solution Sets of Polynomial Systems

Many applications lead to polynomial systems which have positive dimensional solution sets, witnessed by generic points. The set of generic points is partitioned according to the breakup of the solution set into irreducible components. As shown recently in practical computations, this breakup can be predicted efficiently with monodromy. Starting from this predicted partition, points sampled from any component are arranged into a structured grid, and then used to interpolate polynomials vanishing on the solution component. The structured grid allows to construct the interpolating polynomials with divided differences. Applying symmetric functions we obtain an efficient and numerically stable decomposition method. In particular, linear traces suffice to confirm the breakup predicted by monodromy. As linear polynomials are tolerant to roundoff, the decomposition algorithm gains in speed and reliability.

This is joint work with Andrew Sommese and Charles Wampler.