A combinatorial formula for the coefficients of multidimensional resultants

TL;DR

This paper derives an explicit combinatorial formula for the coefficients of multidimensional resultants, using extremal transversals and determinantal inequalities, which clarifies their integer nature and recovers known bounds.

Contribution

It introduces a novel combinatorial approach using extremal transversals to explicitly compute resultant coefficients, advancing algebraic elimination theory.

Findings

Derived an explicit formula for multidimensional resultant coefficients

Showed the coefficients are integer-valued

Recovered Sombra's height bound for resultants

Abstract

The classical multidimensional resultant can be defined as the, suitably normalized, generator of a projective elimination ideal in the ring of universal coefficients. This is the approach via the so-called inertia forms or Tr\"{a}gheitsformen. Using clever substitutions, Mertens and Hurwitz gave a criterion, for recognizing such inertia forms, which amounts to a linear system for their numerical coefficients. In this article we explicitly solve this linear system. We do so by identifying a subset of the available equations which forms a unitriangular system. The key notion we use is that of transversal, i.e., a selection of a monomial term in each of the homogeneous polynomials at hand. We need two such transversals which are disjoint and extremal, in the sense that they relate to extremizers of a, possibly new, determinantal inequality for differences of two substochastic matrices. Thanks to this notion of extremal pair of transversals, we derive an explicit formula for the coefficients of general multidimensional resultants, as a sum of terms made of a sign times a product of multinomial coefficients, thereby explicitly showing they are integer-valued. As an application of our formula, we recover Sombra's bound on the height of resultants, in the classical case.