Bilinear systems with two supports: Koszul resultant matrices, eigenvalues, and eigenvectors

TL;DR

This paper develops a method to compute the multihomogeneous resultant of bilinear systems with different supports using Koszul matrices, extending eigenvalue techniques to solve such algebraic systems.

Contribution

It introduces a constructive approach to express the resultant as a determinant of a Koszul matrix for mixed bilinear systems and extends eigenvalue methods to this setting.

Findings

Resultant expressed as determinant of Koszul matrix

Algorithm for solving mixed bilinear systems

Extended eigenvalues and eigenvectors criterion

Abstract

A fundamental problem in computational algebraic geometry is the computation of the resultant. A central question is when and how to compute it as the determinant of a matrix. whose elements are the coefficients of the input polynomials up-to sign. This problem is well understood for unmixed multihomogeneous systems, that is for systems consisting of multihomogeneous polynomials with the * 1 same support. However, little is known for mixed systems, that is for systems consisting of polynomials with different supports. We consider the computation of the multihomogeneous resultant of bilinear systems involving two different supports. We present a constructive approach that expresses the resultant as the exact determinant of a Koszul resultant matrix, that is a matrix constructed from maps in the Koszul complex. We exploit the resultant matrix to propose an algorithm to solve such systems. In the process we extend the classical eigenvalues and eigenvectors criterion to a more general setting. Our extension of the eigenvalues criterion applies to a general class of matrices, including the Sylvester-type and the Koszul-type ones.