Article

Conditions for determinantal formula for resultant of a polynomial system

Authors:

Arthur D. Chtcherba,

Deepak KapurAuthors Info & Claims

ISSAC '06: Proceedings of the 2006 international symposium on Symbolic and algebraic computation

Pages 55 - 62

https://doi.org/10.1145/1145768.1145784

Published: 09 July 2006 Publication History

Abstract

Matrices constructed from a parameterized multivariate polynomial system are analyzed to ensure that such a matrix contains a condition for the polynomial system to have common solutions irrespective of whether its parameters are specialized or not. Such matrices include resultant matrices constructed using well-known methods for computing resultants over projective, toric and affine varieties. Conditions on these matrices are identified under which the determinant of a maximal minor of such a matrix is a nontrivial multiple of the resultant over a given variety. This condition on matrices allows a generalization of a linear algebra construction, called rank submatrix, for extracting resultants from singular resultant matrices, as proposed by Kapur, Saxena and Yang in ISSAC'94. This construction has been found crucial for computing resultants of non-generic, specialized multivariate polynomial systems that arise in practical applications. The new condition makes the rank submatrix construction based on maximal minor more widely applicable by not requiring that the singular resultant matrix have a column independent of the remaining columns. Unlike perturbation methods, which require introducing a new variable, rank submatrix construction is faster and effective. Properties and conditions on symbolic matrices constructed from a polynomial system are discussed so that the resultant can be computed as a factor of the determinant of a maximal non-singular submatrix.

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Cited By

Dumas JKaltofen EVillard GZhi LYap CEl Din M(2017)Polynomial Time Interactive Proofs for Linear Algebra with Exponential Matrix Dimensions and Scalars Given by Polynomial Time CircuitsProceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3087604.3087640(125-132)Online publication date: 23-Jul-2017
https://dl.acm.org/doi/10.1145/3087604.3087640
Minimair M(2017)Computing the Dixon Resultant with the Maple Package DRApplications of Computer Algebra10.1007/978-3-319-56932-1_19(273-287)Online publication date: 27-Jul-2017
https://doi.org/10.1007/978-3-319-56932-1_19
Kapur DMinimair M(2011)Multivariate Resultants in Bernstein BasisAutomated Deduction in Geometry10.1007/978-3-642-21046-4_4(60-85)Online publication date: 2011
https://doi.org/10.1007/978-3-642-21046-4_4
Show More Cited By

Index Terms

Conditions for determinantal formula for resultant of a polynomial system
1. Mathematics of computing
  1. Mathematical software

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cover image ACM Conferences

ISSAC '06: Proceedings of the 2006 international symposium on Symbolic and algebraic computation

July 2006

374 pages

ISBN:1595932763

DOI:10.1145/1145768

General Chair:
Barry Trager
IBM T.J. Watson Research Center, USA

Copyright © 2006 ACM.

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Published: 09 July 2006

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ISSAC06: International Symposium on Symbolic and Algebraic Computation

July 9 - 12, 2006

Genoa, Italy

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Cited By

Dumas JKaltofen EVillard GZhi LYap CEl Din M(2017)Polynomial Time Interactive Proofs for Linear Algebra with Exponential Matrix Dimensions and Scalars Given by Polynomial Time CircuitsProceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3087604.3087640(125-132)Online publication date: 23-Jul-2017
https://dl.acm.org/doi/10.1145/3087604.3087640
Minimair M(2017)Computing the Dixon Resultant with the Maple Package DRApplications of Computer Algebra10.1007/978-3-319-56932-1_19(273-287)Online publication date: 27-Jul-2017
https://doi.org/10.1007/978-3-319-56932-1_19
Kapur DMinimair M(2011)Multivariate Resultants in Bernstein BasisAutomated Deduction in Geometry10.1007/978-3-642-21046-4_4(60-85)Online publication date: 2011
https://doi.org/10.1007/978-3-642-21046-4_4
Kapur DMinimair M(2008)Multivariate resultants in Bernstein basisProceedings of the 7th international conference on Automated deduction in geometry10.5555/2008257.2008262(60-85)Online publication date: 22-Sep-2008
https://dl.acm.org/doi/10.5555/2008257.2008262

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