Subresultants of Several Univariate Polynomials in Newton Basis
Weidong Wang, Jing Yang
TL;DR
The paper addresses the problem of formulating subresultants for several univariate polynomials expressed in Newton basis while preserving the basis and matching the power-basis results after expansion. It introduces a companion-matrix framework in Newton basis and a generalized determinant-polynomial representation to express these subresultants, connecting root-, coefficient-, and determinant-formulations. A main result provides an explicit construction of a matrix $N_{\lambda,\delta}(F)$ from which the $\,\delta$-th subresultant $S_{\delta}(F)$ can be computed as a determinant polynomial in the Newton basis, thereby enabling a basis-preserving gcd computation without basis transformations. The approach generalizes Barnett-type subresultants to multi-polynomial systems and highlights the flexibility of Newton-basis representations, with open questions on node selection and computational efficiency.
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.