Koszul-type determinantal formulas for families of mixed multilinear systems

TL;DR

This paper develops new Koszul-type determinantal formulas for mixed multilinear polynomial systems, enabling efficient computation of resultants and solving related eigenvalue problems without explicit resultant calculation.

Contribution

It introduces novel determinantal formulas for mixed multilinear systems using Weyman complexes, extending known formulas to new classes of polynomial systems.

Findings

Constructed determinantal formulas for systems with polynomials differing in one block of variables.

Extended Sylvester-type formulas to mixed multilinear systems using Koszul complexes.

Enabled solving polynomial systems via associated matrices without explicit resultant computation.

Abstract

Effective computation of resultants is a central problem in elimination theory and polynomial system solving. Commonly, we compute the resultant as a quotient of determinants of matrices and we say that there exists a determinantal formula when we can express it as a determinant of a matrix whose elements are the coefficients of the input polynomials. We study the resultant in the context of mixed multilinear polynomial systems, that is multilinear systems with polynomials having different supports, on which determinantal formulas were not known. We construct determinantal formulas for two kind of multilinear systems related to the Multiparameter Eigenvalue Problem (MEP): first, when the polynomials agree in all but one block of variables; second, when the polynomials are bilinear with different supports, related to a bipartite graph. We use the Weyman complex to construct Koszul-type determinantal formulas that generalize Sylvester-type formulas. We can use the matrices associated to these formulas to solve square systems without computing the resultant. The combination of the resultant matrices with the eigenvalue and eigenvector criterion for polynomial systems leads to a new approach for solving MEP.