Sylvester double sums, subresultants and symmetric multivariate Hermite interpolation

Abstract

Sylvester doubles sums, introduced first by Sylvester (see Sylvester, 1840, 1853), are symmetric expressions of the roots of two polynomials P and Q. Sylvester's definition of double sums makes no sense if P and Q have multiple roots, since the definition involves denominators that vanish when there are multiple roots. The aims of this paper are to give a new definition for Sylvester double sums making sense if P and Q have multiple roots, which coincides with the definition by Sylvester in the case of simple roots, to prove the fundamental property of Sylvester double sums, i.e. that Sylvester double sums indexed by (k,ℓ) are equal up to a constant if they share the same value for k+ℓ, and to prove 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 (see Lascoux and Pragacz, 2002; d'Andrea et al., 2007; Roy and Szpirglas, 2011). The more general proofs given here are using generalized Vandermonde determinants and a new symmetric multivariate Hermite interpolation as well as an induction on the length of the remainder sequence of P and Q.