2021/05/26 by Matías R. Bender, Jean‐Charles Faugère, Bender, Matías R. +5
Computer Science · Mathematics · #13P15 (Primary) 14Q20 15A18 #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Computer and information sciences #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Polynomial and algebraic computation #Symbolic Computation (cs.SC)
paper · pdf · doi:10.48550/arxiv.2105.13188
openalex publication_date 2021/05/26 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
Effective computation of resultants is a central problem in elimination\ntheory and polynomial system solving. Commonly, we compute the resultant as a\nquotient of determinants of matrices and we say that there exists a\ndeterminantal formula when we can express it as a determinant of a matrix whose\nelements are the coefficients of the input polynomials. We study the resultant\nin the context of mixed multilinear polynomial systems, that is multilinear\nsystems with polynomials having different supports, on which determinantal\nformulas were not known. We construct determinantal formulas for two kind of\nmultilinear systems related to the Multiparameter Eigenvalue Problem (MEP):\nfirst, when the polynomials agree in all but one block of variables; second,\nwhen the polynomials are bilinear with different supports, related to a\nbipartite graph. We use the Weyman complex to construct Koszul-type\ndeterminantal formulas that generalize Sylvester-type formulas. We can use the\nmatrices associated to these formulas to solve square systems without computing\nthe resultant. The combination of the resultant matrices with the eigenvalue\nand eigenvector criterion for polynomial systems leads to a new approach for\nsolving MEP.\n