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Double Sylvester sums for subresultants and multiSchur functions
 J. Symbolic Comput
"... Abstract 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 ..."
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Abstract 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 multiSchur functions and divided differences.
An algorithm for computing the complete root classification of a parametric polynomial
 Lecture Notes in Computer Science 4120
, 2006
"... An improved algorithm, together with its implementation, is presented for the automatic computation of the complete root classification of a real parametric polynomial. The algorithm offers improved efficiency and a new test for nonrealizable conditions. The improvement lies in the direct use of ‘s ..."
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Cited by 3 (2 self)
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An improved algorithm, together with its implementation, is presented for the automatic computation of the complete root classification of a real parametric polynomial. The algorithm offers improved efficiency and a new test for nonrealizable conditions. The improvement lies in the direct use of ‘sign lists’, obtained from the discriminant sequence, rather than ‘revised sign lists’. It is shown that the discriminant sequences, upon which the sign lists are based, are closely related both to SturmHabicht sequences and to subresultant sequences. Thus calculations based on any of these quantities are essentially equivalent. One particular application of complete root classifications is the determination of the conditions for the positive definiteness of a polynomial, and here the new algorithm is applied to a class of sparse polynomials. It is seen that the number of conditions for positive definiteness remains surprisingly small in these cases. Key words: complete root classification, parametric polynomial, real quantifier elimination, real root, subresultant polynomials. 1.
Recursive Polynomial Remainder Sequence and the Nested Subresultants
, 806
"... Abstract. We give two new expressions of subresultants, nested subresultant and reduced nested subresultant, for the recursive polynomial remainder sequence (PRS) which has been introduced by the author. The reduced nested subresultant reduces the size of the subresultant matrix drastically compared ..."
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Abstract. We give two new expressions of subresultants, nested subresultant and reduced nested subresultant, for the recursive polynomial remainder sequence (PRS) which has been introduced by the author. The reduced nested subresultant reduces the size of the subresultant matrix drastically compared with the recursive subresultant proposed by the authors before, hence it is much more useful for investigation of the recursive PRS. Finally, we discuss usage of the reduced nested subresultant in approximate algebraic computation, which motivates the present work. 1
Abstract
, 806
"... We introduce concepts of “recursive polynomial remainder sequence (PRS) ” and “recursive subresultant, ” along with investigation of their properties. A recursive PRS is defined as, if there exists the GCD (greatest common divisor) of initial polynomials, a sequence of PRSs calculated “recursively ” ..."
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We introduce concepts of “recursive polynomial remainder sequence (PRS) ” and “recursive subresultant, ” along with investigation of their properties. A recursive PRS is defined as, if there exists the GCD (greatest common divisor) of initial polynomials, a sequence of PRSs calculated “recursively ” for the GCD and its derivative until a constant is derived, and recursive subresultants are defined by determinants representing the coefficients in recursive PRS as functions of coefficients of initial polynomials. We give three different constructions of subresultant matrices for recursive subresultants; while the first one is builtup just with previously defined matrices thus the size of the matrix increases fast as the recursion deepens, the last one reduces the size of the matrix drastically by the Gaussian elimination on the second one which has a “nested ” expression, i.e. a Sylvester matrix whose elements are themselves determinants. Key words: polynomial remainder sequence, subresultants, Gaussian elimination, Sylvester’s identity 1
Bezout matrices, Subresultants and Parameters
"... The problem of computing the greatest common divisor of two univariate polynomials in D[x], with D a field or an integral domain, is one of the cornerstones in Computer Algebra. When D is an integral domain we shall talk about a greatest common divisor as a polynomial in D[x] which is a greatest com ..."
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The problem of computing the greatest common divisor of two univariate polynomials in D[x], with D a field or an integral domain, is one of the cornerstones in Computer Algebra. When D is an integral domain we shall talk about a greatest common divisor as a polynomial in D[x] which is a greatest common divisor of the
Article Submitted to Journal of Symbolic Computation 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 multiSchur functions and divided differences. In this paper, we provide an ..."
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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 multiSchur functions and divided differences. In this paper, we provide an elementary proof that uses only basic properties of matrix multiplication and Vandermonde determinants.
PARDI!
 ISAAC
, 2001
"... We propose a new algorithm for converting a characteristic set of a prime differential ideal from one ranking into another. This differential algebra algorithm computes characteristic sets by change of ranking (ordering) for prime ideals. It identifies the purely algebraic subproblems which arise d ..."
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We propose a new algorithm for converting a characteristic set of a prime differential ideal from one ranking into another. This differential algebra algorithm computes characteristic sets by change of ranking (ordering) for prime ideals. It identifies the purely algebraic subproblems which arise during differential computations and solves them algebraically. There are two improvements w.r.t. other approaches: formerly unsolved problems could be carried out; it is conceptually simple. Different variants are implemented.
An Elementary Proof of Sylvester’s Double Sums for Subresultants
, 2006
"... 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 multiSchur functions and divided differences. In this paper, we provide an ..."
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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 multiSchur functions and divided differences. In this paper, we provide an elementary proof that uses only basic properties of matrix multiplication and Vandermonde determinants.
A Simple Proof of Sylvester’s Double Sums for Subresultants
, 2006
"... In 1853 Sylvester stated an elegant formula that expresses the subresultants in terms of the roots of the input polynomials. The validity of this formula was recently independently proved by Apéry and Jouanolou and by Lascoux and Pragacz, by using the theory of multiSchur functions. In this paper w ..."
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In 1853 Sylvester stated an elegant formula that expresses the subresultants in terms of the roots of the input polynomials. The validity of this formula was recently independently proved by Apéry and Jouanolou and by Lascoux and Pragacz, by using the theory of multiSchur functions. In this paper we provide a simpler proof that uses only basic properties of matrix multiplication and Vandermonde determinants.