Sylvester's double sums: an inductive proof of the general case (Q414627)

Let \(A\) and \(B\) be two set of indeterminants. For every pair of integers \(p\) and \(q\), such that \(0\leq p\leq |A|\) and \(0\leq q\leq |B|\), in 1853 Sylvester defined the following polynomial in the variable \(x\) called the double sum expression in \(A\) and \(B\):NEWLINENEWLINENEWLINE\[NEWLINE\mathrm{Syl}^{p,q}(A,B):=\sum_{A'\subset A, |A'|=p;\,\,B'\subset B, |B'|=q}R(x,A')R(x,B')\frac{R(A',B')R(A\setminus A',B\setminus B')}{R(A',A\setminus A')R(B',B\setminus B')},NEWLINE\]NEWLINENEWLINENEWLINEwhere for two sets of indeterminants \(X\) and \(Y\) one has NEWLINE\[NEWLINER(X,Y)=\Pi_{y\in Y,x\in X}(y-x).NEWLINE\]NEWLINENEWLINENEWLINEFor two polynomials NEWLINE\[NEWLINEf=\Pi_{\alpha\in A}(x-\alpha),NEWLINE\]NEWLINE NEWLINE\[NEWLINEg=\Pi_{\beta\in B}(x-\beta)NEWLINE\]NEWLINE one can consider the \(k\)-th subresultant \(\mathrm{Sres}_k(f,g)\). The subresultant is a polynomial in \(x\), it is defined as a determinant of a matrix, whose elements are either coefficients of \(f\) and \(g\) or polynomials \(f\) and \(g\) multiplied by some power of \(x\).NEWLINENEWLINEAs it was pointed out by Sylvester in 1853, double sums can be expressed through \(f\) and \(g\) and the subresultants of \(f\) and \(g\). However for some \(p\) and \(q\) he did not give explicit expressions. They were given by \textit{C. D'Andrea, H. Hong, T. Krick} and \textit{A. Szanto} [``Sylvester's double sums: the general case'', J. Symb. Comput. 44, No. 9, 1164--1175 (2009; Zbl 1217.13013)] .NEWLINENEWLINELater \textit{M.-F. Roy} and \textit{A. Szpirglas} [``Sylvester double sums and subresultants'', J. Symb. Comput. 46, No. 4, 385--395 (2011; Zbl 1213.12003)] had given in some cases an elementary proof of this result.NEWLINENEWLINEThe main result of the paper under review is a new elementary proof of these results in all cases. As the authors say, the main motivation is to explore and tackle the problem concerning of the definition of Sylvester's double sums to the case of multiple roots and their connection to subresultants.