Sparse trace tests

TL;DR

This paper develops numerical trace tests for sparse polynomial systems, extending classical methods by analyzing sparse resultants and monodromy groups to verify solution completeness.

Contribution

It introduces new trace test algorithms tailored for sparse systems, leveraging structural analysis of sparse resultants and monodromy group extensions.

Findings

New numerical tests for solution completeness in sparse systems

Extension of classical trace test to sparse polynomial systems

Analysis of sparse resultants and monodromy groups

Abstract

We establish how the coefficients of a sparse polynomial system influence the sum (or the trace) of its zeros. As an application, we develop numerical tests for verifying whether a set of solutions to a sparse system is complete. These algorithms extend the classical trace test in numerical algebraic geometry. Our results rely on both the analysis of the structure of sparse resultants as well as an extension of Esterov's results on monodromy groups of sparse systems.