Article

Solving Kaltofen's challenge on Zolotarev's approximation problem

Author:

Daniel LazardAuthors Info & Claims

ISSAC '06: Proceedings of the 2006 international symposium on Symbolic and algebraic computation

Pages 196 - 203

https://doi.org/10.1145/1145768.1145803

Published: 09 July 2006 Publication History

Get Access

Abstract

Zolotarev's Approximation problem consists in finding the best approximation of a polynomial of degree n by a polynomial of degree n−2 in the interval [−1, 1]. It is solved for n ≤ 12 by replacing CAD (Cylindrical Algebraic Decomposition) by a strategy consisting in computing first the Zariski closure of the semi-algebraic set which is looked for, before managing the inequalities.

References

[1]

N. I. Achieser. Theory of approximation. Translated by Charles J. Hyman. Frederick Ungar Publishing Co., New York, 1956.

Google Scholar

[2]

George E. Collins. Application of Quantifier Elimination to Solotareff's Approximation Problem. Technical Report 95-31, RISC Report Series, University of Linz, Austria, 1995.

Google Scholar

[3]

Michael Kalkbrener. A generalized Euclidean algorithm for computing triangular representations of algebraic varieties. J. Symbolic Comput., 15(2):143--167, 1993.

Digital Library

Google Scholar

[4]

Erich Kaltofen, Robert M. Corless, and David J. Jeffrey. Challenges of symbolic computation: my favorite open problems. J. Symbolic Comput., 29(6):891--919, 2000.

Digital Library

Google Scholar

[5]

Daniel Lazard and Fabrice Rouillier. Solving parametric polynomial systems. J. Symbolic Comput., 2006. Under revision, Research Report RR-5149, INRIA, 2004.

Google Scholar

Cited By

View all

Akbari FHashemi A(2024)On Saturation of Zero-Dimensional IdealsExperimental Mathematics10.1080/10586458.2024.2423179(1-12)Online publication date: 13-Nov-2024
https://doi.org/10.1080/10586458.2024.2423179
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
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

Solving Kaltofen's challenge on Zolotarev's approximation problem
1. Applied computing
  1. Physical sciences and engineering
    1. Mathematics and statistics
2. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Computer algebra systems
    2. Symbolic and algebraic algorithms
      1. Algebraic algorithms

Recommendations

The Distribution of Zeros and Poles of Asymptotically Extremal Rational Functions for Zolotarev's Problem

We investigate the possible limit distributions of zeros and poles associated with ray sequences of rational functions that are asymptotically optimal for weighted Zolotarev problems. For disjoint compacta E"1, E"2 in the complex plane, the Zolotarev ...
The two-point Padé approximation problem and its Hankel vector
Abstract
The two-point Padé approximation problem is to find a ratio of two coprime polynomials with some constraints on their degrees to approximate a function whose power series expansions at the origin and at infinity are given. In this paper, we ...
Zolotarev ω-Polynomials inWrH^ω[0,1]

The main result of this paper characterizes generalizationsof Zolotarev polynomials as extremal functions in the Kolmogorov Landauproblemf(m)(0) sup,f WrH 0,1], f C0,1] B, (( ))where (t) is a concave modulus of continuity,r,m:1 m r,are integers, andB B0(...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

ISSAC '06: Proceedings of the 2006 international symposium on Symbolic and algebraic computation

July 2006

374 pages

ISBN:1595932763

DOI:10.1145/1145768

General Chair:
Barry Trager
IBM T.J. Watson Research Center, USA

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]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 09 July 2006

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Qualifiers

Article

Conference

ISSAC06

Sponsor:

ISSAC06: International Symposium on Symbolic and Algebraic Computation

July 9 - 12, 2006

Genoa, Italy

Acceptance Rates

Overall Acceptance Rate 395 of 838 submissions, 47%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

6
Total Citations
View Citations
136
Total Downloads

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

Reflects downloads up to 20 Nov 2024

Other Metrics

View Author Metrics

Citations

Cited By

View all

Akbari FHashemi A(2024)On Saturation of Zero-Dimensional IdealsExperimental Mathematics10.1080/10586458.2024.2423179(1-12)Online publication date: 13-Nov-2024
https://doi.org/10.1080/10586458.2024.2423179
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
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
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
Grasegger GVo NYap CEl Din M(2017)An Algebraic-Geometric Method for Computing Zolotarev PolynomialsProceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation10.1145/3087604.3087613(173-180)Online publication date: 23-Jul-2017
https://dl.acm.org/doi/10.1145/3087604.3087613
Strzeboński AKoepf W(2010)Computation with semialgebraic sets represented by cylindrical algebraic formulasProceedings of the 2010 International Symposium on Symbolic and Algebraic Computation10.1145/1837934.1837952(61-68)Online publication date: 25-Jul-2010
https://dl.acm.org/doi/10.1145/1837934.1837952

View Options

Login options

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

Cited By

Index Terms

Recommendations

The Distribution of Zeros and Poles of Asymptotically Extremal Rational Functions for Zolotarev's Problem

The two-point Padé approximation problem and its Hankel vector

Zolotarev ω-Polynomials inWrH^ω[0,1]