Continuation of root functionals of a system of polynomial equations and the reduction of polynomials modulo its ideal

TL;DR

This paper introduces the concept of continuation of root functionals and polynomial reduction modulo ideals, extending the theory of root functionals for systems of polynomial equations with finite solutions.

Contribution

It develops the operation of continuation of root functionals and polynomial reduction based on extension of bounded root functionals for 0-dimensional polynomial ideals.

Findings

Defined the operation of continuation of root functionals.

Connected extension operation with multivariate Bezoutian construction.

Provided a framework for polynomial reduction modulo ideals.

Abstract

The notion of a root functional of polynomials is a generalization of the notion of a root for a multiple root. A root functional is a linear functional that is defined on a polynomial ring and annuls the ideal of a system of polynomials. A bounded root functional is a functional that annuls d-th component of the ideal in some filtration in this ideal. It was constructed the operation of continuation of root functionals and the operation of reduction of polynomials modulo the ideal on the basis of the extension operation for bounded root functionals when the number of polynomials is equal to the number of variables and the ideal of polynomials is 0-dimensional. The extension operation has connection with the multivariate Bezoutian construction.