research-article

Multihomogeneous resultant formulae for systems with scaled support

Authors:

Ioannis Z. Emiris,

Angelos A. MantzaflarisAuthors Info & Claims

ISSAC '09: Proceedings of the 2009 international symposium on Symbolic and algebraic computation

Pages 143 - 150

https://doi.org/10.1145/1576702.1576724

Published: 28 July 2009 Publication History

Abstract

Constructive methods for matrices of multihomogeneous resultants for unmixed systems have been studied in [7, 13, 15]. We generalize these constructions to mixed systems, whose Newton polytopes are scaled copies of one polytope, thus taking a step towards systems with arbitrary supports. First, we specify matrices whose determinant equals the resultant and characterize the systems that admit such formulae. Bézout-type determinantal formulae do not exist, but we describe all possible Sylvester-type and hybrid formulae. We establish tight bounds for the corresponding degree vectors, as well as precise domains where these concentrate; the latter are new even for the unmixed case. Second, we make use of multiplication tables and strong duality theory to specify resultant matrices explicitly, in the general case. The encountered matrices are classified; these include a new type of Sylvester-type matrix as well as Bézout-type matrices, which we call partial Bezoutians. Our public-domain Maple implementation includes efficient storage of complexes in memory, and construction of resultant matrices.

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Cited By

Emiris IAbramov SZima EGao X(2016)Compact Formulae in Sparse EliminationProceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2930889.2930943(1-4)Online publication date: 20-Jul-2016
https://dl.acm.org/doi/10.1145/2930889.2930943
Tsigaridas E(2014)Algebraic Algorithms*Computing Handbook, Third Edition10.1201/b16812-13(1-30)Online publication date: 8-May-2014
https://doi.org/10.1201/b16812-13

Index Terms

Multihomogeneous resultant formulae for systems with scaled support
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Algebraic algorithms

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cover image ACM Conferences

ISSAC '09: Proceedings of the 2009 international symposium on Symbolic and algebraic computation

July 2009

402 pages

ISBN:9781605586090

DOI:10.1145/1576702

General Chairs:
Jeremy Johnson
Drexel University, USA
,
Hyungju Park
Korea Institute for Advanced Study, Korea
,
Program Chair:
Erich Kaltofen
North Carolina State University, USA

Copyright © 2009 ACM.

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Published: 28 July 2009

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ISSAC '09: International Symposium on Symbolic and Algebraic Computation

July 28 - 31, 2009

Seoul, Republic of Korea

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Cited By

Emiris IAbramov SZima EGao X(2016)Compact Formulae in Sparse EliminationProceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2930889.2930943(1-4)Online publication date: 20-Jul-2016
https://dl.acm.org/doi/10.1145/2930889.2930943
Tsigaridas E(2014)Algebraic Algorithms*Computing Handbook, Third Edition10.1201/b16812-13(1-30)Online publication date: 8-May-2014
https://doi.org/10.1201/b16812-13

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