research-article

On the Bit Complexity of Solving Bilinear Polynomial Systems

Authors:

Ioannis Z. Emiris,

Angelos Mantzaflaris,

Elias TsigaridasAuthors Info & Claims

ISSAC '16: Proceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation

Pages 215 - 222

https://doi.org/10.1145/2930889.2930919

Published: 20 July 2016 Publication History

Abstract

We bound the Boolean complexity of computing isolating hyperboxes for all complex roots of systems of bilinear polynomials. The resultant of such systems admits a family of determinantal Sylvester-type formulas, which we make explicit by means of homological complexes. The computation of the determinant of the resultant matrix is a bottleneck for the overall complexity. We exploit the quasi-Toeplitz structure to reduce the problem to efficient matrix-vector products, corresponding to multivariate polynomial multiplication. For zero-dimensional systems, we arrive at a primitive element and a rational univariate representation of the roots. The overall bit complexity of our probabilistic algorithm is OB(n⁴ D⁴ + n²D⁴ τ), where n is the number of variables, D equals the bilinear Bezout bound, and τ is the maximum coefficient bitsize. Finally, a careful infinitesimal symbolic perturbation of the system allows us to treat degenerate and positive dimensional systems, thus making our algorithms and complexity analysis applicable to the general case.

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Cited By

Bender MFaugère JMantzaflaris ATsigaridas EKauers MOvchinnikov ASchost É(2018)Bilinear Systems with Two SupportsProceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3208976.3209011(63-70)Online publication date: 11-Jul-2018
https://dl.acm.org/doi/10.1145/3208976.3209011
Mantzaflaris ASchost ETsigaridas EYap CEl Din M(2017)Sparse Rational Univariate RepresentationProceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3087604.3087653(301-308)Online publication date: 23-Jul-2017
https://dl.acm.org/doi/10.1145/3087604.3087653
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

Index Terms

On the Bit Complexity of Solving Bilinear Polynomial Systems
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Symbolic calculus algorithms

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cover image ACM Conferences

ISSAC '16: Proceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation

July 2016

434 pages

ISBN:9781450343800

DOI:10.1145/2930889

General Chairs:
Sergei Abramov
Russian Academy of Sciences, Russia
,
Eugene Zima
Wilfrid Laurier University, Canada
,
Program Chair:
Xiao-Shan Gao
Chinese Academy of Sciences, China

Copyright © 2016 ACM.

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Published: 20 July 2016

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ISSAC '16: International Symposium on Symbolic and Algebraic Computation

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Cited By

Bender MFaugère JMantzaflaris ATsigaridas EKauers MOvchinnikov ASchost É(2018)Bilinear Systems with Two SupportsProceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3208976.3209011(63-70)Online publication date: 11-Jul-2018
https://dl.acm.org/doi/10.1145/3208976.3209011
Mantzaflaris ASchost ETsigaridas EYap CEl Din M(2017)Sparse Rational Univariate RepresentationProceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3087604.3087653(301-308)Online publication date: 23-Jul-2017
https://dl.acm.org/doi/10.1145/3087604.3087653
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

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