Determination of the basis of the space of all root functionals of a system of polynomial equations and of the basis of its ideal by the operation of the extension of bounded root functionals

TL;DR

This paper presents an algorithm to find bases of the ideal and root functionals space for 0-dimensional polynomial systems using extension operations, with complexity tied to polynomial degrees and variables.

Contribution

It introduces a novel algorithm leveraging extension operations of bounded root functionals to determine bases of ideals and root functionals for polynomial systems.

Findings

Algorithm computes bases with complexity d^{O(n)}

Extension operation relates to multivariate Bezoutian

Applicable to 0-dimensional polynomial ideals

Abstract

It is proposed the algorithm that find a basis of the ideal and a basis of the space of all root functionals by using the extension operation for bounded root functionals, when the number of polynomials is equal to the number of variables, if it is known that the ideal of polynomials is 0-dimensional. The asyptotic complexity of this algorithm is d^{O(n)} operations, where n is the number of polynomials and the number of variables, d is the maximal degree of polynomials. The extension operation has connection with the multivariate Bezoutian construction.