Sylvester's double sums: an inductive proof of the general case

TL;DR

This paper provides an inductive proof to fully describe Sylvester's double sums, connecting them to polynomial subresultants and extending Sylvester's original work from 1853.

Contribution

It introduces an inductive method to derive the complete form of Sylvester's double sums, building on previous partial descriptions and proofs.

Findings

Complete description of Sylvester's double sums achieved

Inductive proof method successfully applied to polynomial expressions

Links between double sums and polynomial subresultants clarified

Abstract

In 1853 J. Sylvester introduced a family of double sum expressions for two finite sets of indeterminates and showed that some members of the family are essentially the polynomial subresultants of the monic polynomials associated with these sets. In 2009, in a joint work with C. D'Andrea and H. Hong we gave the complete description of all the members of the family as expressions in the coefficients of these polynomials. In 2010, M.-F. Roy and A. Szpirglas presented a new and natural inductive proof for the cases considered by Sylvester. Here we show how induction also allows to obtain the full description of Sylvester's double-sums.