Sylvester's double sums: an inductive proof of the general case
type: Item
instance of: scholarly article
title: Sylvester's double sums: an inductive proof of the general case
zbMATH Open document ID: 1244.15004
author: Teresa Krick
author: Agnes Szanto
published in: Journal of Symbolic Computation
publication date: 2012-05-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/1106.4770
Mathematics Subject Classification ID: 13P15
Mathematics Subject Classification ID: 68W30
Mathematics Subject Classification ID: 15A15
zbMATH DE Number: 6033269
zbMATH Keywords: Sylvester's double sums
zbMATH Keywords: subresultants
zbMATH Keywords: polynomial
zbMATH Keywords: determinant
reviewed by: Dmitry Artamonov
MaRDI profile type: Publication
OpenAlex ID: W2020399563
Wikidata QID: Q57431033
arXiv ID: 1106.4770
cites work: Subresultants and Reduced Polynomial Remainder Sequences
cites work: An elementary proof of Sylvester's double sums for subresultants
cites work: Sylvester's double sums: the general case
cites work: Q4406533
cites work: Q4023676
cites work: Sylvester subresultants, rational Cauchy approximations, Thiele's continued fractions and higher Bruhat orders
cites work: Double Sylvester sums for subresultants and multi-Schur functions.
cites work: Sylvester double sums and subresultants
DOI: 10.1016/J.JSC.2012.01.003
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review text: 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 t…
description: scientific article; zbMATH DE number 6033269