research-article

Rational univariate representations of bivariate systems and applications

Authors:

Yacine Bouzidi,

Sylvain Lazard,

Fabrice RouillierAuthors Info & Claims

ISSAC '13: Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation

Pages 109 - 116

https://doi.org/10.1145/2465506.2465519

Published: 26 June 2013 Publication History

Abstract

We address the problem of solving systems of two bivariate polynomials of total degree at most d with integer coefficients of maximum bitsize τ We suppose known a linear separating form (that is a linear combination of the variables that takes different values at distinct solutions of the system) and focus on the computation of a Rational Univariate Representation (RUR).

We present an algorithm for computing a RUR with worst-case bit complexity in Õ_B(d⁷+d⁶τ) and bound the bitsize of its coefficients by Õ(d²+dτ) (where Õ_B refers to bit complexities and Õ to complexities where polylogarithmic factors are omitted). We show in addition that isolating boxes of the solutions of the system can be computed from the RUR with Õ_B(d⁸+d⁷τ) bit operations. Finally, we show how a RUR can be used to evaluate the sign of a bivariate polynomial (of degree at most d and bitsize at most τ) at one real solution of the system in Õ_B(d⁸+d⁷τ) bit operations and at all the ϴ(d²) solutions in only O(d) times that for one solution.

References

[1]

M.-E. Alonso, E. Becker, M.-F. Roy, and T. Wörmann. Multiplicities and idempotents for zerodimensional systems. In Algorithms in Algebraic Geometry and Applications, vol. 143 of Progress in Mathematics, pp. 1--20, 1996.

Digital Library

[2]

S. Basu, R. Pollack, and M.-F. Roy. Algorithms in Real Algebraic Geometry, volume 10 of Algorithms and Computation in Mathematics. Springer-Verlag, 2006.

Digital Library

[3]

Y. Bouzidi, S. Lazard, M. Pouget, and F. Rouillier. Rational Univariate Representations of Bivariate Systems and Applications. Research Report RR-8262, INRIA, 2013.

Digital Library

[4]

Y. Bouzidi, S. Lazard, M. Pouget, and F. Rouillier. Separating linear forms for bivariate systems. In Proc. ISSAC, 2013.

Digital Library

[5]

L. Busé, H. Khalil, and B. Mourrain. Resultant-based methods for plane curves intersection problems. In Computer Algebra in Scientific Computing (CASC), vol. 3718 of LNCS, pp. 75--92, 2005.

Digital Library

[6]

J. Canny. A new algebraic method for robot motion planning and real geometry. In Proc. FoCS, pp. 39--48, 1987.

Digital Library

[7]

J. Cheng, S. Lazard, L. Peñaranda, M. Pouget, F. Rouillier, and E. Tsigaridas. On the topology of real algebraic plane curves. Mathematics in Computer Science, 4:113--137, 2010.

[8]

D. I. Diochnos, I. Z. Emiris, and E. P. Tsigaridas. On the asymptotic and practical complexity of solving bivariate systems over the reals. J. Symb. Comput., 44(7):818--835, 2009.

Digital Library

[9]

P. Emeliyanenko and M. Sagraloff. On the complexity of solving a bivariate polynomial system. In Proc. ISSAC, pp. 154--161, 2012.

Digital Library

[10]

W. Fulton. Algebraic curves: an introduction to algebraic geometry. 2008.% Personal reprint made available by the author http://www.math.lsa.umich.edu/ wfulton/CurveBook.pdf

[11]

L. Gonz�lez-Vega and M. El Kahoui. An improved upper complexity bound for the topology computation of a real algebraic plane curve. J. of Complexity, 12(4):527--544, 1996.

Digital Library

[12]

T. Lickteig and M.-F. Roy. Sylvester-Habicht Sequences and Fast Cauchy Index Computation. J. Symb. Comput., 31(3):315--341, 2001.

Digital Library

[13]

M. Mignotte. Math�matiques pour le calcul formel. Presses Universitaires de France, 1989.

[14]

V. Y. Pan. Solving a polynomial equation: some history and recent progress. SIAM Review, 39(2):187--220, 1997.

Digital Library

[15]

F. Rouillier. Solving zero-dimensional systems through the rational univariate representation. J. of Applicable Algebra in Engineering, Communication and Computing, 9(5):433--461, 1999.

[16]

S. M. Rump. Polynomial minimum root separation. Mathematics of Computation, 33(145):327--336, 1979.

[17]

M. Sagraloff. When Newton meets Descartes: A Simple and Fast Algorithm to Isolate the Real Roots of a Polynomial. In Proc. ISSAC, pp.297--304 2012.

Digital Library

[18]

E. Schost. Sur la R�solution des Syst�mes Polynomiaux � Param�tres. PhD thesis, Ecole Polytechnique, France, 2001.

[19]

A. Sch�nhage. The fundamental theorem of algebra in terms of computational complexity. Manuscript, 1982.

[20]

J. von zur Gathen and J. Gerhard. Modern Computer Algebra. Cambridge Univ. Press, Cambridge, U.K., 2003.

Digital Library

[21]

C. Yap. Fundamental Problems of Algorithmic Algebra. Oxford University Press, Oxford-New York, 2000.

Digital Library

Cited By

Schost �St-Pierre C(2024) An -adic algorithm for bivariate Gr�bner 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
Katsamaki CRouillier FTsigaridas EZafeirakopoulos Z(2023) PTOPOJournal of Symbolic Computation10.1016/j.jsc.2022.08.012115:C(427-451)Online publication date: 1-Mar-2023
https://dl.acm.org/doi/10.1016/j.jsc.2022.08.012
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
Show More Cited By

Index Terms

Rational univariate representations of bivariate systems and applications
1. Theory of computation
  1. Design and analysis of algorithms

Recommendations

Separating linear forms and Rational Univariate Representations of bivariate systems

We address the problem of solving systems of bivariate polynomials with integer coefficients. We first present an algorithm for computing a separating linear form of such systems, that is a linear combination of the variables that takes different values ...
Rational Univariate Representation of Zero-Dimensional Ideals with Parameters
ISSAC '22: Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation

An algorithm for computing the rational univariate representation of zero-dimensional ideals with parameters is presented in the paper. Different from the rational univariate representation of zero-dimensional ideals without parameters, the number of ...
Solving bivariate systems using Rational Univariate Representations

Given two coprime polynomials P and Q in Z x , y of degree bounded by d and bitsize bounded by ź , we address the problem of solving the system { P , Q } . We are interested in certified numerical approximations or, more precisely, isolating boxes of ...

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

ISSAC '13: Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation

June 2013

400 pages

ISBN:9781450320597

DOI:10.1145/2465506

General Chairs:
Michael Monagan
Simon Fraser University, Burnaby, Canada
,
Gene Cooperman
Northeastern University, Boston, USA
,
Program Chair:
Mark Giesbrecht
University of Waterloo, Waterloo, Canada

Copyright � 2013 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]

Sponsors

SIGSAM: ACM Special Interest Group on Symbolic and Algebraic Manipulation

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 26 June 2013

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Conference

ISSAC'13

Sponsor:

SIGSAM

ISSAC'13: International Symposium on Symbolic and Algebraic Computation

June 26 - 29, 2013

Maine, Boston, USA

Acceptance Rates

Overall Acceptance Rate 395 of 838 submissions, 47%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

10
Total Citations
View Citations
88
Total Downloads

Downloads (Last 12 months)4
Downloads (Last 6 weeks)0

Reflects downloads up to 17 Oct 2024

Other Metrics

View Author Metrics

Citations

Cited By

Schost �St-Pierre C(2024) An -adic algorithm for bivariate Gr�bner 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
Katsamaki CRouillier FTsigaridas EZafeirakopoulos Z(2023) PTOPOJournal of Symbolic Computation10.1016/j.jsc.2022.08.012115:C(427-451)Online publication date: 1-Mar-2023
https://dl.acm.org/doi/10.1016/j.jsc.2022.08.012
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
Katsamaki CRouillier FTsigaridas EZafeirakopoulos ZEmiris IZhi LMantzaflaris A(2020)On the geometry and the topology of parametric curvesProceedings of the 45th International Symposium on Symbolic and Algebraic Computation10.1145/3373207.3404062(281-288)Online publication date: 20-Jul-2020
https://dl.acm.org/doi/10.1145/3373207.3404062
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
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
Kobel ASagraloff M(2015)On the complexity of computing with planar algebraic curvesJournal of Complexity10.1016/j.jco.2014.08.00231:2(206-236)Online publication date: Apr-2015
https://doi.org/10.1016/j.jco.2014.08.002
Sagraloff MNagasaka KWinkler FSzanto A(2014)A near-optimal algorithm for computing real roots of sparse polynomialsProceedings of the 39th International Symposium on Symbolic and Algebraic Computation10.1145/2608628.2608632(359-366)Online publication date: 23-Jul-2014
https://dl.acm.org/doi/10.1145/2608628.2608632
Falbel EKoseleff PRouillier F(2014)Representations of fundamental groups of 3-manifolds into $$\mathrm{PGL}(3,\mathbb {C})$$ PGL ( 3 , C ) : exact computations in low complexityGeometriae Dedicata10.1007/s10711-014-9987-x177:1(229-255)Online publication date: 22-May-2014
https://doi.org/10.1007/s10711-014-9987-x
Cheng JJin K(2014)Finding a Deterministic Generic Position for an Algebraic Space CurveComputer Algebra in Scientific Computing10.1007/978-3-319-10515-4_6(74-84)Online publication date: 2014
https://doi.org/10.1007/978-3-319-10515-4_6

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents