Multihomogeneous Resultant Formulae for Systems with Scaled Support

TL;DR

This paper extends the construction of multihomogeneous resultants to scaled mixed systems, providing explicit matrices, bounds, and classifications, including new Sylvester-type and Bezout-type matrices, with a Maple implementation.

Contribution

It generalizes existing methods to scaled mixed systems, characterizes systems with resultant matrices, and introduces new matrix types and bounds, advancing the computational algebraic geometry field.

Findings

Characterization of systems admitting Sylvester-type and hybrid formulae

Explicit construction of resultant matrices for scaled systems

Introduction of new Sylvester-type and partial Bezoutian matrices

Abstract

Constructive methods for matrices of multihomogeneous (or multigraded) resultants for unmixed systems have been studied by Weyman, Zelevinsky, Sturmfels, Dickenstein and Emiris. We generalize these constructions to mixed systems, whose Newton polytopes are scaled copies of one polytope, thus taking a step towards systems with arbitrary supports. First, we specify matrices whose determinant equals the resultant and characterize the systems that admit such formulae. Bezout-type determinantal formulae do not exist, but we describe all possible Sylvester-type and hybrid formulae. We establish tight bounds for all corresponding degree vectors, and specify domains that will surely contain such vectors; the latter are new even for the unmixed case. Second, we make use of multiplication tables and strong duality theory to specify resultant matrices explicitly, for a general scaled system, thus including unmixed systems. The encountered matrices are classified; these include a new type of Sylvester-type matrix as well as Bezout-type matrices, known as partial Bezoutians. Our public-domain Maple implementation includes efficient storage of complexes in memory, and construction of resultant matrices.