Toric Sylvester forms

TL;DR

This paper introduces toric Sylvester forms to analyze the saturation of ideals in toric varieties, providing new tools for elimination theory and polynomial system solving.

Contribution

It establishes a duality property and explicitly constructs toric Sylvester forms, offering new bases for graded components of ideal quotients and applications in elimination theory.

Findings

Toric Sylvester forms form bases for certain ideal quotients.

New elimination matrices are introduced for solving sparse polynomial systems.

A determinant expression of the sparse resultant is derived from a modified Koszul complex.

Abstract

In this paper, we investigate the structure of the saturation of ideals generated by sparse homogeneous polynomials over a projective toric variety $X$ with respect to the irrelevant ideal of $X$. As our main results, we establish a duality property and make it explicit by introducing toric Sylvester forms, under a certain positivity assumption on $X$. In particular, we prove that toric Sylvester forms yield bases of some graded components of $I^{\text{sat}}/I$, where $I$ denotes an ideal generated by $n+1$ generic forms, $n$ is the dimension of $X$ and $I^{\text{sat}}$ the saturation of $I$ with respect to the irrelevant ideal of the Cox ring of $X$. Then, to illustrate the relevance of toric Sylvester forms we provide three consequences in elimination theory over smooth toric varieties: (1) we introduce a new family of elimination matrices that can be used to solve sparse polynomial systems by means of linear algebra methods, including overdetermined polynomial systems; (2) by incorporating toric Sylvester forms to the classical Koszul complex associated to a polynomial system, we obtain new expressions of the sparse resultant as a determinant of a complex; (3) we explote a new formula for computing toric residues of the product of two forms.