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 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.     

Ore Subresultant Coefficients in Solutions

The subresultants play a fundamental role in elimination theory and computer algebra. Recently they have been extended to Ore polynomials. They are defined by an expression in the coefficients of Ore polynomials. In this paper, we provide another expression for them. This expression is written in terms of the solutions of Ore polynomials (in generic case). It is a generalization of our previous paper where we gave expressions for the principal (formal leading) coefficients of subresultants. In this paper, we give expressions for all the other coefficients. Received: September 3, 2000     

Sylvester double sums and subresultants

Sylvester double sums, introduced first by Sylvester (see [Sylvester, 1840] and [Sylvester, 1853]), are symmetric expressions of the roots of two polynomials, while subresultants are defined through the coefficients of these polynomials (see Apery and Jouanolou (2006) and Basu et al. (2003) for references on subresultants). As pointed out by Sylvester, the two notions are very closely related: Sylvester double sums and subresultants are equal up to a multiplicative non-zero constant in the ground field. Two proofs are already known: that of Lascoux and Pragacz (2003), using Schur functions, and that of d’Andrea et al. (2007), using manipulations of matrices. The purpose of this paper is to give a new simple proof using similar inductive properties of double sums and subresultants.     

Sylvester’s double sums: The general case

In 1853 Sylvester introduced a family of double-sum expressions for two finite sets of indeterminates and showed that some members of the family are essentially the polynomial subresultants of the monic polynomials associated with these sets. A question naturally arises: What are the other members of the family? This paper provides a complete answer to this question. The technique that we developed to answer the question turns out to be general enough to characterize all members of the family, providing a uniform method.     

An algebraic approach to continuous collision detection for ellipsoids

We present algebraic expressions for characterizing three configurations formed by two ellipsoids in R³ that are relevant to collision detection: separation, external touching and overlapping. These conditions are given in terms of explicit formulae expressed by the subresultant sequence of the characteristic polynomial of the two ellipsoids and its derivative. For any two ellipsoids, the signs of these formulae can easily be evaluated to classify their configuration. Furthermore, based on these algebraic conditions, an efficient method is developed for continuous collision detection of two moving ellipsoids under arbitrary motions.     

Ore Principal Subresultant Coefficients in Solutions

The principal subresultant coefficients of polynomials play a fundamental role in elimination theory and computer algebra. Recently they have been extended to Ore polynomials. They are defined by an expression in the coefficients of Ore polynomials. In this paper, we provide another expression for them. This expression is written in terms of the “solutions” of Ore polynomials (in “generic” case). It can be viewed as a generalization of the well known expression for resultants of two commutative polynomials: the product of the pair-wise differences of their roots. Received: August 16, 1999; revised version: July 3, 2000