An Elementary Proof of Sylvester's Double Sums for Subresultants
  In 1853 Sylvester stated and proved an elegant formula that expresses the polynomial subresultants in terms of the roots of the input polynomials. Sylvester's formula was also recently proved by Lascoux and Pragacz by using multi-Schur functions and divided differences. In this paper, we provide an elementary proof that uses only basic properties of matrix multiplication and Vandermonde determinants. 
Carlos D'Andrea, Hoon Hong, Teresa Krick, Agnes Szanto