research-article

On the complexity of solving a bivariate polynomial system

Authors:

Pavel Emeliyanenko,

Michael SagraloffAuthors Info & Claims

ISSAC '12: Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation

Pages 154 - 161

https://doi.org/10.1145/2442829.2442854

Published: 22 July 2012 Publication History

Abstract

We study the complexity of computing the real solutions of a bivariate polynomial system using the recently presented algorithm Bisolve [2]. Bisolve is an elimination method which, in a first step, projects the solutions of a system onto the x- and y-axes and, then, selects the actual solutions from the so induced candidate set. However, unlike similar algorithms, Bisolve requires no genericity assumption on the input, and there is no need for any kind of coordinate transformation. Furthermore, extensive benchmarks as presented in [2] confirm that the algorithm is highly practical, that is, a corresponding C++ implementation in Cgal outperforms state of the art approaches by a large factor. In this paper, we focus on the theoretical complexity of Bisolve. For two polynomials f, g ∈ Z[x, y] of total degree at most n with integer coefficients bounded by 2^τ, we show that Bisolve computes isolating boxes for all real solutions of the system f = g = 0 using O(n⁸ + n⁷τ) bit operations, thereby improving the previous record bound for the same task by several magnitudes.

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Cited By

Batra PSharma V(2024)Complexity of a Root Clustering Algorithm for Holomorphic FunctionsTheoretical Computer Science10.1016/j.tcs.2024.114504(114504)Online publication date: Mar-2024
https://doi.org/10.1016/j.tcs.2024.114504
Schost ÉSt-Pierre C(2024) An -adic algorithm for bivariate Gröbner bases Journal of Symbolic Computation10.1016/j.jsc.2024.102389(102389)Online publication date: Sep-2024
https://doi.org/10.1016/j.jsc.2024.102389
Schost ÉSt-Pierre C(2024)Newton iteration for lexicographic Gröbner bases in two variablesJournal of Algebra10.1016/j.jalgebra.2024.04.018Online publication date: Apr-2024
https://doi.org/10.1016/j.jalgebra.2024.04.018
Show More Cited By

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On the complexity of solving a bivariate polynomial system

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cover image ACM Other conferences

ISSAC '12: Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation

July 2012

390 pages

ISBN:9781450312691

DOI:10.1145/2442829

General Chair:
Joris van der Hoeven
École Polytechnique, France
,
Program Chair:
Mark van Hoeij
Florida State U.

Copyright © 2012 ACM.

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Grenoble University: Grenoble University
INRIA: Institut Natl de Recherche en Info et en Automatique

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SIGSAM: ACM Special Interest Group on Symbolic and Algebraic Manipulation

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Association for Computing Machinery

New York, NY, United States

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Published: 22 July 2012

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ISSAC'12

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Grenoble University
INRIA

ISSAC'12: International Symposium on Symbolic and Algebraic Computation

July 22 - 25, 2012

Grenoble, France

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ISSAC '12 Paper Acceptance Rate 46 of 86 submissions, 53%;

Overall Acceptance Rate 395 of 838 submissions, 47%

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Cited By

Batra PSharma V(2024)Complexity of a Root Clustering Algorithm for Holomorphic FunctionsTheoretical Computer Science10.1016/j.tcs.2024.114504(114504)Online publication date: Mar-2024
https://doi.org/10.1016/j.tcs.2024.114504
Schost ÉSt-Pierre C(2024) An -adic algorithm for bivariate Gröbner bases Journal of Symbolic Computation10.1016/j.jsc.2024.102389(102389)Online publication date: Sep-2024
https://doi.org/10.1016/j.jsc.2024.102389
Schost ÉSt-Pierre C(2024)Newton iteration for lexicographic Gröbner bases in two variablesJournal of Algebra10.1016/j.jalgebra.2024.04.018Online publication date: Apr-2024
https://doi.org/10.1016/j.jalgebra.2024.04.018
Jin KCheng J(2021)On the Complexity of Computing the Topology of Real Algebraic Space CurvesJournal of Systems Science and Complexity10.1007/s11424-020-9164-2Online publication date: 12-Jan-2021
https://doi.org/10.1007/s11424-020-9164-2
Consolini LLocatelli M(2017)On the complexity of quadratic programming with two quadratic constraintsMathematical Programming: Series A and B10.1007/s10107-016-1073-8164:1-2(91-128)Online publication date: 1-Jul-2017
https://dl.acm.org/doi/10.1007/s10107-016-1073-8
Brand CSagraloff MAbramov SZima EGao X(2016)On the Complexity of Solving Zero-Dimensional Polynomial Systems via ProjectionProceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2930889.2930934(151-158)Online publication date: 20-Jul-2016
https://dl.acm.org/doi/10.1145/2930889.2930934
Moroz GSchost EAbramov SZima EGao X(2016)A Fast Algorithm for Computing the Truncated ResultantProceedings of the 2016 ACM International Symposium on Symbolic and Algebraic Computation10.1145/2930889.2930931(341-348)Online publication date: 20-Jul-2016
https://dl.acm.org/doi/10.1145/2930889.2930931
Burr M(2016)Continuous amortization and extensionsJournal of Symbolic Computation10.1016/j.jsc.2016.01.00777:C(78-126)Online publication date: 1-Nov-2016
https://dl.acm.org/doi/10.1016/j.jsc.2016.01.007
Pan VTsigaridas E(2016)Nearly optimal refinement of real roots of a univariate polynomialJournal of Symbolic Computation10.1016/j.jsc.2015.06.00974:C(181-204)Online publication date: 1-May-2016
https://dl.acm.org/doi/10.1016/j.jsc.2015.06.009
Cheng JJin K(2015)A generic position based method for real root isolation of zero-dimensional polynomial systemsJournal of Symbolic Computation10.1016/j.jsc.2014.09.01768:P1(204-224)Online publication date: 1-May-2015
https://dl.acm.org/doi/10.1016/j.jsc.2014.09.017
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