Multigraded Sylvester forms, Duality and Elimination Matrices

TL;DR

This paper introduces multigraded Sylvester forms and duality properties to derive new elimination matrices, advancing methods for solving multihomogeneous polynomial systems in multiprojective spaces.

Contribution

It generalizes duality and Sylvester form concepts to multihomogeneous systems, leading to novel elimination matrices for multiprojective polynomial solving.

Findings

Established a duality property for multihomogeneous systems

Explicitly constructed multigraded Sylvester forms

Derived new elimination matrices for solving polynomial systems

Abstract

In this paper we study the equations of the elimination ideal associated with $n+1$ generic multihomogeneous polynomials defined over a product of projective spaces of dimension $n$. We first prove a duality property and then make this duality explicit by introducing multigraded Sylvester forms. These results provide a partial generalization of similar properties that are known in the setting of homogeneous polynomial systems defined over a single projective space. As an important consequence, we derive a new family of elimination matrices that can be used for solving zero-dimensional multiprojective polynomial systems by means of linear algebra methods.