Degeneracy loci and polynomial equation solving

TL;DR

This paper introduces a new probabilistic pseudo-polynomial time algorithm for solving polynomial equations over complex varieties, leveraging degeneracy loci and polar varieties for improved computational efficiency.

Contribution

It develops a novel algorithm based on degeneracy loci and polar varieties to solve polynomial equations efficiently over the reals and affine spaces.

Findings

Algorithm achieves bounded error probabilistic complexity

Effective for polynomial equation solving over reals

Applicable to computing generic fibers of endomorphisms

Abstract

Let V be a smooth equidimensional quasi-affine variety of dimension r over the complex numbers $C$ and let $F$ be a $(p\times s)$-matrix of coordinate functions of $C[V]$, where $s\ge p+r$. The pair $(V,F)$ determines a vector bundle $E$ of rank $s-p$ over $W:=\{x\in V:\mathrm{rk} F(x)=p\}$. We associate with $(V,F)$ a descending chain of degeneracy loci of E (the generic polar varieties of $V$ represent a typical example of this situation). The maximal degree of these degeneracy loci constitutes the essential ingredient for the uniform, bounded error probabilistic pseudo-polynomial time algorithm which we are going to design and which solves a series of computational elimination problems that can be formulated in this framework. We describe applications to polynomial equation solving over the reals and to the computation of a generic fiber of a dominant endomorphism of an affine space.