Computing all Affine Solution Sets of Binomial Systems

TL;DR

This paper introduces a novel method for computing all affine solution sets of binomial systems by exploiting Newton polytope structures, effectively handling solutions with zero components and outperforming existing numerical algebraic geometry methods on benchmarks.

Contribution

The paper presents a new enumeration-based approach for affine solutions of binomial systems that accounts for zero components, improving scalability over traditional methods.

Findings

Method efficiently computes affine solutions including zero components.

Outperforms witness set methods on benchmark problems.

Scales better for large sparse polynomial systems.

Abstract

To compute solutions of sparse polynomial systems efficiently we have to exploit the structure of their Newton polytopes. While the application of polyhedral methods naturally excludes solutions with zero components, an irreducible decomposition of a variety is typically understood in affine space, including also those components with zero coordinates. For the problem of computing solution sets in the intersection of some coordinate planes, the direct application of a polyhedral method fails, because the original facial structure of the Newton polytopes may alter completely when selected variables become zero. Our new proposed method enumerates all factors contributing to a generalized permanent and toric solutions as a special case of this enumeration. For benchmark problems such as the adjacent 2-by-2 minors of a general matrix, our methods scale much better than the witness set representations of numerical algebraic geometry.