Symmetric Subresultants and Applications

TL;DR

This paper introduces symmetric sub-resultants of polynomials, providing a structure theorem, a fast Euclidean division-based algorithm, and a new fraction-free Toeplitz matrix inversion method suitable for computer algebra.

Contribution

It generalizes Schur's transforms through symmetric sub-resultants, establishes their structure, and develops efficient algorithms for their computation and Toeplitz matrix inversion.

Findings

Fast algorithm for symmetric sub-resultants using Euclidean division and FFT

Deep link established between symmetric sub-resultants and Toeplitz matrices

Fraction-free Toeplitz matrix inversion algorithm suitable for computer algebra

Abstract

Schur's transforms of a polynomial are used to count its roots in the unit disk. These are generalized them by introducing the sequence of symmetric sub-resultants of two polynomials. Although they do have a determinantal definition, we show that they satisfy a structure theorem which allows us to compute them with a type of Euclidean division. As a consequence, a fast algorithm based on a dichotomic process and FFT is designed. We prove also that these symmetric sub-resultants have a deep link with Toeplitz matrices. Finally, we propose a new algorithm of inversion for such matrices. It has the same cost as those already known, however it is fraction-free and consequently well adapted to computer algebra.