Closed formula for univariate subresultants in multiple roots

TL;DR

This paper presents a novel closed-form formula for univariate subresultants expressed in terms of roots, applicable to polynomials with arbitrary multiplicities, extending previous special-case results.

Contribution

It generalizes Sylvester sums to multisets and provides the first universal formula for subresultants based solely on roots, regardless of multiplicity structure.

Findings

Derived a closed formula for subresultants in terms of roots

Extended Sylvester sums to multisets with repeated elements

Utilized multivariate symmetric interpolation and Exchange Lemma

Abstract

We generalize Sylvester single sums to multisets (sets with repeated elements), and show that these sums compute subresultants of two univariate polyomials as a function of their roots independently of their multiplicity structure. This is the first closed formula for subresultants in terms of roots that works for arbitrary polynomials, previous efforts only handled special cases. Our extension involves in some cases confluent Schur polynomials, and is obtained by using multivariate symmetric interpolation via an Exchange Lemma.