research-article

An Algebraic-Geometric Method for Computing Zolotarev Polynomials

Authors:

Georg Grasegger,

N. Thieu VoAuthors Info & Claims

ISSAC '17: Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation

Pages 173 - 180

https://doi.org/10.1145/3087604.3087613

Published: 23 July 2017 Publication History

Abstract

In this paper we study a differential equation which arises from the theory of Zolotarev polynomials. By extending a symbolic algorithm for finding rational solutions of algebraic ordinary differential equations, we construct a method for computing explicit expressions for Zolotarev polynomials. This method is an algebraic geometric one and works subject to (radical) parametrization of algebraic curves. As a main application we compute the explicit form of the proper Zolotarev polynomial of degree 5.

References

[1]

2016. NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov, Release 1.0.13 of 2016-09--16. (2016). F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller and B. V. Saunders, eds.

[2]

N. I. Achieser. 1992. Theory of Approximation. Dover Publications, New York.

[3]

G. Chen and Y. Ma. 2005. Algorithmic reduction and rational general solutions of first order algebraic differential equations. In Differential equations with symbolic computation. Birkhäuser, Basel, 201--212.

[4]

G. E. Collins. 1995. Application of Quantifier Elimination to Solotareff's Approximation Problem. RISC Report Series 1995--31. Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz.

[5]

R. Feng and X.-S. Gao. 2004. Rational General Solutions of Algebraic Ordinary Differential Equations. In Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation, J. Gutierrez (Ed.). ACM, New York, 155--162.

Digital Library

[6]

L. Fuchs. 1884. Über Differentialgleichungen, deren Integrale feste Verzwei- gungspunkte besitzen. 1884 (1884), 699--710.

[7]

M. Harrison. 2013. Explicit solution by radicals, gonal maps and plane models of algebraic curves of genus 5 or 6. Journal of Symbolic Computation 51 (2013), 3--21.

Digital Library

[8]

E. Kaltofen. 2000. Challenges of Symbolic Computation: My Favorite Open Problems. Journal of Symbolic Computation 29, 6 (2000), 891--919.

Digital Library

[9]

J. J. Kovacic. 1986. An algorithm for solving second order linear homogeneous differential equations. Journal of Symbolic Computation 2, 1 (1986), 3--43.

Digital Library

[10]

D. Lazard. 2006. Solving Kaltofen's Challenge on Zolotarev's Approximation Problem. In Proceedings of the 2006 International Symposium on Symbolic and Algebraic Computation. ACM, New York, 196--203.

Digital Library

[11]

V. I. Lebedev. 1994. Zolotarev polynomials and extremum problems. Russian Journal of Numerical Analysis and Mathematical Modelling 9, 3 (1994), 231--263.

[12]

M. van Hoeij. 1995. An Algorithm for Computing the Weierstrass Normal Form. In Proceedings of the 1995 International Symposium on Symbolic and Algebraic Computation. ACM, New York, 90--95.

Digital Library

[13]

M. van Hoeij. 2002. An algorithm for computing the Weierstrass normal form of hyperelliptic curves. Technical Report 0203130.arXiv.org.

[14]

V. A. Malyshev. 2003. Algebraic solution of the Zolotarev problem. St. Petersburg Mathematical Journa 14, 4 (2003), 238--240.

[15]

V. A. Markov. 1916. Über Polynome, die in einem gegebenen Intervalle möglichst wenig von Null abweichen. Mathematische Annalen 77 (1916), 213--258.

[16]

L. X. C. Ngô and F. Winkler. 2010. Rational general solutions of first order non-autonomous parametrizable ODEs. Journal of Symbolic Computation 45, 12 (2010), 1426--1441.

Digital Library

[17]

H.-J. Rack. 1989. On Polynomials with Largest Coefficient Sums. Journal of Approximation Theory 56 (1989), 348--359.

Digital Library

[18]

J. Schicho, F.-O. Schreyer, and M. Weimann. 2013. Computational aspects of gonal maps and radical parametrization of curves. Applicable Algebra in Engineering, Communication and Computing 24, 5 (2013), 313--341.

[19]

J. Schicho and D. Sevilla. 2012. Effective radical parametrization of trigonal curves. In Computational algebraic and analytic geometry. Contemporary Mathematics, Vol. 572. American Mathematical Society, Providence, 221--231.

[20]

K. Schiefermayr. 2007. Inverse polynomial images which consists of two Jordan arcs -- an algebraic solution. Journal of Approximation Theory 148, 2 (2007), 148--157.

Digital Library

[21]

J. R. Sendra, F. Winkler, and S. Pérez-Díaz. 2008. Rational Algebraic Curves, A Computer Algebra Approach. Algorithms and Computation in Mathematics, Vol. 22. Springer, Berlin Heidelberg.

Digital Library

[22]

J. R. Sendra and D. Sevilla. 2011. Radical parametrizations of algebraic curves by adjoint curves. Journal of Symbolic Computation 46, 9 (2011), 1030--1038.

Digital Library

[23]

A. Shadrin. 2004. Twelve proofs of the Markov inequality. In Approximation theory: a volume dedicated to Borislav Bojanov, D. K. Dimitrov, G. Nikolov, and R. Uluchev (Eds.). Prof. Marin Drinov Academic Publishing House, Sofia, 233-- 298.

[24]

M. L. Sodin and P. M. Yuditskij. 1991. Algebraic solution of a problem of E. I. Zolotarev and N. I. Akhiezer on polynomials with smallest deviation from zero. Journal of Mathematical Sciences 76, 4 (1991), 2486--2492.

Cited By

Rack HVajda R(2021)An explicit radical parametrization of Zolotarev polynomials of degree 7 in terms of nested square rootsAdvanced Studies: Euro-Tbilisi Mathematical Journal10.32513/asetmj/193220081314:4Online publication date: 1-Dec-2021
https://doi.org/10.32513/asetmj/1932200813
Kubák JSovka P(2021)A Novel Spectral Zoom Method based on Zolotarev Polynomials2021 International Conference on Applied Electronics (AE)10.23919/AE51540.2021.9542876(1-5)Online publication date: 7-Sep-2021
https://dl.acm.org/doi/10.23919/AE51540.2021.9542876
Rack HVajda R(2019)Explicit algebraic solution of Zolotarev's First Problem for low-degree polynomialsJournal of Numerical Analysis and Approximation Theory10.33993/jnaat482-117348:2(175-201)Online publication date: 31-Dec-2019
https://doi.org/10.33993/jnaat482-1173
Show More Cited By

Index Terms

An Algebraic-Geometric Method for Computing Zolotarev Polynomials
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
2. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
      1. Ordinary differential equations

Recommendations

Rational general solutions of algebraic ordinary differential equations
ISSAC '04: Proceedings of the 2004 international symposium on Symbolic and algebraic computation

We give a necessary and sufficient condition for an algebraic ODE to have a rational type general solution. For an autonomous first order ODE, we give an algorithm to compute a rational general solution if it exists. The algorithm is based on the ...
Computing real inflection points of cubic algebraic curves

Shape modeling using planar cubic algebraic curves calls for computing the real inflection points of these curves since inflection points represents important shape feature. A real inflection point is also required for transforming projectively a planar ...
Radical parametrizations of algebraic curves by adjoint curves

We present algorithms for parametrizing by radicals an irreducible curve, not necessarily plane, when the genus is less than or equal to 4 and the curve is defined over an algebraically closed field of characteristic zero. In addition, we also present ...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image ACM Other conferences

ISSAC '17: Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation

July 2017

466 pages

ISBN:9781450350648

DOI:10.1145/3087604

Editor:
Michael Burr
Clemson University, USA
,
General Chair:
Chee K. Yap
New York University, USA
,
Program Chair:
Mohab Safey El Din
University Pierre et Marie Curie, France

Copyright © 2017 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 the author(s) 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].

In-Cooperation

SIGSAM: ACM Special Interest Group on Symbolic and Algebraic Manipulation

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 23 July 2017

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Author Tags

Qualifiers

Research-article

Conference

ISSAC '17

ISSAC '17: International Symposium on Symbolic and Algebraic Computation

July 23 - 28, 2017

Kaiserslautern, Germany

Acceptance Rates

Overall Acceptance Rate 395 of 838 submissions, 47%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

5
Total Citations
View Citations
74
Total Downloads

Downloads (Last 12 months)1
Downloads (Last 6 weeks)0

Reflects downloads up to 25 Nov 2024

Other Metrics

View Author Metrics

Citations

Cited By

Rack HVajda R(2021)An explicit radical parametrization of Zolotarev polynomials of degree 7 in terms of nested square rootsAdvanced Studies: Euro-Tbilisi Mathematical Journal10.32513/asetmj/193220081314:4Online publication date: 1-Dec-2021
https://doi.org/10.32513/asetmj/1932200813
Kubák JSovka P(2021)A Novel Spectral Zoom Method based on Zolotarev Polynomials2021 International Conference on Applied Electronics (AE)10.23919/AE51540.2021.9542876(1-5)Online publication date: 7-Sep-2021
https://dl.acm.org/doi/10.23919/AE51540.2021.9542876
Rack HVajda R(2019)Explicit algebraic solution of Zolotarev's First Problem for low-degree polynomialsJournal of Numerical Analysis and Approximation Theory10.33993/jnaat482-117348:2(175-201)Online publication date: 31-Dec-2019
https://doi.org/10.33993/jnaat482-1173
Winkler F(2019)The Algebro-Geometric Method for Solving Algebraic Differential Equations — A SurveyJournal of Systems Science and Complexity10.1007/s11424-019-8348-032:1(256-270)Online publication date: 14-Feb-2019
https://doi.org/10.1007/s11424-019-8348-0
Rack H(2017)The second Zolotarev case in the Erdös-Szegö solution to a Markov-type extremal problem of SchurJournal of Numerical Analysis and Approximation Theory10.33993/jnaat461-110046:1(54-77)Online publication date: 21-Sep-2017
https://doi.org/10.33993/jnaat461-1100

View Options

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