Solving a sparse systems using linear algebra

TL;DR

This paper introduces a novel theoretical approach combining toric varieties and linear algebra to efficiently solve sparse systems with finitely many solutions, especially overdetermined ones.

Contribution

It adapts eigenvalue and eigenvector theorems to work with the first Koszul map, providing new tools for solving and counting solutions of sparse systems.

Findings

New eigenvalue-based theorems for sparse systems

Method to count expected solutions

Applicable to overdetermined systems

Abstract

We give a new theoretical tool to solve sparse systems with finitely many solutions. It is based on toric varieties and basic linear algebra; eigenvalues, eigenvectors and coefficient matrices. We adapt Eigenvalue theorem and Eigenvector theorem to work with a canonical rectangular matrix (the first Koszul map) and prove that these new theorems serve to solve overdetermined sparse systems and to count the expected number of solutions.