Sylvester double sums, subresultants and symmetric multivariate Hermite interpolation

TL;DR

This paper redefines Sylvester double sums to handle multiple roots, establishes their relation to subresultants, and employs advanced algebraic tools like generalized Vandermonde determinants and Hermite interpolation.

Contribution

It introduces a new definition of Sylvester double sums valid for multiple roots and proves their key properties using generalized algebraic techniques.

Findings

Double sums are equal up to a constant if they share the same sum of indices.

Double sums are proportional to subresultants.

New definitions extend applicability to polynomials with multiple roots.

Abstract

Sylvester doubles sums, introduced first by Sylvester are symmetric expressions of the roots of two polynomials. Sylvester's definition of double sums makes no sense in the presence of multiple roots, since the definition involves denominators that vanish when there are multiple roots. The aim of this paper is to give a new definition of Sylvester double sums making sense in the presence of multiple roots, which coincides with the definition by Sylvester in the case of simple roots, to prove that double sums indexed by $(k,\ell)$ are equal up to a constant if they share the same value for $k+\ell$, as well a proof of the relationship between double sums and subresultants, i.e. that they are equal up to a constant. In the simple root case, proofs of these properties are already known. The more general proofs given here are using generalized Vandermonde determinants and symmetric multivariate Hermite interpolation as well as an induction on the length of the remainder sequence of $P$ and $Q$.