Symmetric Interpolation, Exchange Lemma and Sylvester Sums

TL;DR

This paper explores symmetric multivariate Lagrange interpolation to derive an Exchange Lemma, providing a new explanation for Sylvester's double sum expressions using subresultants and Bezout coefficients.

Contribution

It introduces an Exchange Lemma derived from symmetric interpolation that clarifies Sylvester's double sum expressions in algebraic resultants.

Findings

Derivation of an Exchange Lemma from symmetric interpolation

Simplified explanation of Sylvester's double sum expressions

Connection between subresultants and Bezout coefficients

Abstract

The theory of symmetric multivariate Lagrange interpolation is a beautiful but rather unknown tool that has many applications. Here we derive from it an Exchange Lemma that allows to explain in a simple and natural way the full description of the double sum expressions introduced by Sylvester in 1853 in terms of subresultants and their Bezout coefficients.