Sylvester double sums and subresultants

Sylvester double sums, introduced first by Sylvester (see Sylvester (1840, 1853)), are symmetric expressions of the roots of two polynomials, while subresultants are defined through the coefficients of these polynomials (see Apery and Jouanolou (2006), and Basu et al. (2003), for references on subresultants). As pointed out by Sylvester, the two notions are very closely related: Sylvester double sums and subresultants are equal up to a multiplicative non-zero constant in the ground field. Two proofs are already known: that of Lascoux and Pragacz (2003), , using Schur functions, and that of d’Andrea et al. (2007), using manipulations of matrices. The purpose of this paper is to give a new simple proof using similar inductive properties of double sums and subresultants.