Double Sylvester sums for subresultants and multi-Schur functions

J. J. 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. Introduction and statement of the main result The subresultants played a fundamental role in the theory of polynomial equations in the 19-th century, cf. e.g. (Sylvester, 1839, 1840, 1853), (Borchardt, 1860), and (Salmon, 1885). Recently they also have found important applications in computer algebra, for example, in devising efficient methods for computing greatest common divisors of two polynomials (Collins, 1967, 1973), (Brown, 1971), and (Brown, Traub, 1971), for carrying out quantifier elimination over complex of real numbers cf. e.g. (Collins, 1975), and also for coding theory (Shen, 1992). They have been also extended to some noncommutative polynomials (Chardin, 1991), (Li, 1998), and (Hong, 2001). Suppose that two polynomials in one variable P (x) = x + α1x m−1 + · · ·+ αm and Q(x) = x + β1x + · · ·+ βn with, say, complex coefficients are given. Then the subresultant of degree d asso-