Subresultants of Several Univariate Polynomials in Newton Basis

TL;DR

This paper introduces a novel matrix-based method to compute subresultant polynomials of multiple univariate polynomials expressed in Newton basis, extending determinantal polynomial concepts from power basis and enabling basis-preserving gcd computation.

Contribution

It develops a new formula for subresultants in Newton basis using a specialized matrix and extends determinantal polynomial concepts from power to Newton basis.

Findings

New matrix-based formula for subresultants in Newton basis

Extension of determinantal polynomial concept to Newton basis

Application in basis-preserving gcd computation

Abstract

In this paper, we consider the problem of formulating the subresultant polynomials for several univariate polynomials in Newton basis. It is required that the resulting subresultant polynomials be expressed in the same Newton basis as that used in the input polynomials. To solve the problem, we devise a particular matrix with the help of the companion matrix of a polynomial in Newton basis. Meanwhile, the concept of determinantal polynomial in power basis for formulating subresultant polynomials is extended to that in Newton basis. It is proved that the generalized determinantal polynomial of the specially designed matrix provides a new formula for the subresultant polynomial in Newton basis, which is equivalent to the subresultant polynomial in power basis. Furthermore, we show an application of the new formula in devising a basis-preserving method for computing the gcd of several Newton polynomials.