Article

Sylvester-resultants for bivariate polynomials with planar newton polygons

Authors:

Ron GoldmanAuthors Info & Claims

ISSAC '04: Proceedings of the 2004 international symposium on Symbolic and algebraic computation

Pages 205 - 212

https://doi.org/10.1145/1005285.1005316

Published: 04 July 2004 Publication History

Abstract

We derive necessary and sufficient conditions which guarantee that a multiplying set of monomials generates exactly a Sylvester A-resultant for three bivariate polynomials with a given planar Newton polygon. We show that valid multiplying sets come in complementary pairs, and any two complementary pairs of multiplying sets can be used to index the rows and columns of a pure Bezoutian A-resultant for the same Newton polygon.The necessary and sufficient conditions include a set of Diophantine equations that can be solved to generate the multiplying sets and therefore the corresponding Sylvester $A$-resultants. Examples relevant to Geometric Modeling are provided, including a new family of hexagonal examples for which Sylvester formulas were not previously known. These examples not only flesh out the theory, but also demonstrate that none of the conditions are superfluous and that all the conditions are mutually independent. The proof of the main theorem makes use of tools from algebraic geometry, including sheaf cohomology on toric varieties and Weyman's resultant complex.

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Cited By

Busé LMantzaflaris ATsigaridas E(2019)Matrix formulæ for resultants and discriminants of bivariate tensor-product polynomialsJournal of Symbolic Computation10.1016/j.jsc.2019.07.007Online publication date: Jul-2019
https://doi.org/10.1016/j.jsc.2019.07.007
Mantzaflaris ATsigaridas EYap CEl Din M(2017)Resultants and Discriminants for Bivariate Tensor-Product PolynomialsProceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3087604.3087646(309-316)Online publication date: 23-Jul-2017
https://dl.acm.org/doi/10.1145/3087604.3087646
Emiris IKonaxis C(2011)Single-lifting Macaulay-type formulae of generalized unmixed sparse resultantsJournal of Symbolic Computation10.1016/j.jsc.2011.02.00246:8(919-942)Online publication date: 1-Aug-2011
https://dl.acm.org/doi/10.1016/j.jsc.2011.02.002
Show More Cited By

Index Terms

Sylvester-resultants for bivariate polynomials with planar newton polygons
1. Computing methodologies
  1. Symbolic and algebraic manipulation

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cover image ACM Conferences

ISSAC '04: Proceedings of the 2004 international symposium on Symbolic and algebraic computation

July 2004

334 pages

ISBN:158113827X

DOI:10.1145/1005285

General Chair:
Josef Schicho
RICAM, Austrian Academy of Sciences, Linz, Austria

Copyright © 2004 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Published: 04 July 2004

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July 4 - 7, 2004

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Cited By

Busé LMantzaflaris ATsigaridas E(2019)Matrix formulæ for resultants and discriminants of bivariate tensor-product polynomialsJournal of Symbolic Computation10.1016/j.jsc.2019.07.007Online publication date: Jul-2019
https://doi.org/10.1016/j.jsc.2019.07.007
Mantzaflaris ATsigaridas EYap CEl Din M(2017)Resultants and Discriminants for Bivariate Tensor-Product PolynomialsProceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3087604.3087646(309-316)Online publication date: 23-Jul-2017
https://dl.acm.org/doi/10.1145/3087604.3087646
Emiris IKonaxis C(2011)Single-lifting Macaulay-type formulae of generalized unmixed sparse resultantsJournal of Symbolic Computation10.1016/j.jsc.2011.02.00246:8(919-942)Online publication date: 1-Aug-2011
https://dl.acm.org/doi/10.1016/j.jsc.2011.02.002
Kaltofen EKoiran PSendra JGonzalez-Vega L(2008)Expressing a fraction of two determinants as a determinantProceedings of the twenty-first international symposium on Symbolic and algebraic computation10.1145/1390768.1390790(141-146)Online publication date: 20-Jul-2008
https://dl.acm.org/doi/10.1145/1390768.1390790

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