Extension of bounded root functionals of a system of polynomial equations

TL;DR

This paper explores bounded root functionals of polynomial systems, extending their properties and linking them to multivariate Bezoutian constructions, especially for systems with equal numbers of equations and unknowns.

Contribution

It introduces an extension operation for bounded root functionals of polynomial systems, connecting it with multivariate Bezoutian, for systems where the number of equations equals the number of unknowns.

Findings

Extension operation for bounded root functionals is developed.

Connection established between root functionals and multivariate Bezoutian.

Applicable to systems with equal number of equations and unknowns.

Abstract

The notion of a root functional of a system of polynomials or ideal of polynomials is a generalization of the notion of a root, in particular, for a multiple root. A root functional is a linear functional that is defined on a polynomial ring and annuls the ideal of polynomials. A bounded root functional is a functional that annuls d-th component of the ideal in some filtration in this ideal. The paper consider bounded root functionals and their extension operation for a system of polynomial equation at which the number of equations is equal to the number of unknows. The extension operation has connection with the multivariate Bezoutian construction.