Double Sylvester sums for subresultants and multi-Schur functions

Sylvester has announced formulas expressing the subresultants (or the successive polynomial remainders for the Euclidean division) of two polynomials, in terms of some double sums over the roots of the two polynomials. We prove Sylvester formulas using the techniques of multivariate polynomials involving multi-Schur functions and divided differences.