Article

Free access

The Chinese remainder problem, multivariate interpolation, and Gröbner bases

Authors:

Thomas Becker,

Volker WeispfenningAuthors Info & Claims

ISSAC '91: Proceedings of the 1991 international symposium on Symbolic and algebraic computation

Pages 64 - 69

https://doi.org/10.1145/120694.120703

Published: 01 June 1991 Publication History

PDF eReader

References

[1]

L.J. Billera and L.L. Rose, GrSbner basis methods for multivariate splines, Rutcors Research Report 1-89 (1989)

Google Scholar

[2]

B. Buchberger, GrSbner bases: an algorithmic method in polynomial ideal theory, in: N.K. Bose (ed.), Progress, Directions and Open Problems in Multidimendional Systems Theory, pp. 184-232, D. Riedel Publ. Comp. (Dordrecht 1985)

Google Scholar

[3]

P.R. Cappello, E. Gallopoulos, and ~.K. Ko~, Systolic computation of interpolating polynomials, Computing 45, 95-117 (1990)

Digital Library

Google Scholar

[4]

P.G. Ciarlet and P.A. Raviart, General Lagrange and Hermite interpolation in R" with applications to finite element methods, Arch. Rational Mech. Anal. 46, 177-199, (1972)

Crossref

Google Scholar

[5]

M. Gasca and J.I. Maeztu, On Lagrange and Hermite interpolation in Rk, Numer. Math. 39, 1-14 (1982)

Digital Library

Google Scholar

[6]

R.B. Guenther and E.L. Roetman, Some observations on interpolation in higher dimensions, Mathematics of Computation, Volume 24, Number 111 (1970)

Crossref

Google Scholar

[7]

H.M. MSller, Mehrdimensionale Hermite-interpolation und numerische Integration, Math. Z. 148, 107-118 (1976)

Crossref

Google Scholar

[8]

H.C. Thatcher Jr. and W.E. Milne, Interpolation in several variables, J. Soc. Indust. Appl. Math. 8, 33-42 (1960)

Crossref

Google Scholar

[9]

H. Werner, Remarks on Newton type multivariate interpolation for subsets of grids, Computing 25, 181-191 (1980)

Crossref

Google Scholar

[10]

O. Zariski and P. Samuel, Commutative Algebra Volume I, Springer Graduate Texts in Mathematics (Berlin 1979)

Google Scholar

Cited By

View all

Curtis PMaruri-Aguilar H(2016)Design for Smooth Models over Complex RegionsmODa 11 - Advances in Model-Oriented Design and Analysis10.1007/978-3-319-31266-8_9(71-78)Online publication date: 7-Jun-2016
https://doi.org/10.1007/978-3-319-31266-8_9
DERBISZ J(2014)REMARKS ON MULTIVARIATE EXTENSIONS OF POLYNOMIAL BASED SECRET SHARING SCHEMESNational Security Studies10.37055/sbn/1352386:2(285-297)Online publication date: 5-Dec-2014
https://doi.org/10.37055/sbn/135238
Montes AWibmer M(2010)Gröbner bases for polynomial systems with parametersJournal of Symbolic Computation10.1016/j.jsc.2010.06.01745:12(1391-1425)Online publication date: 1-Dec-2010
https://dl.acm.org/doi/10.1016/j.jsc.2010.06.017
Show More Cited By

Index Terms

The Chinese remainder problem, multivariate interpolation, and Gröbner bases
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Algebraic algorithms
2. Mathematics of computing
  1. Discrete mathematics
  2. Mathematical analysis
    1. Numerical analysis
      1. Computations on polynomials
      2. Number-theoretic computations

Recommendations

Chinese Remainder Theorem for bivariate lexicographic Gröbner bases
ISSAC '23: Proceedings of the 2023 International Symposium on Symbolic and Algebraic Computation

This article proposes an algorithm that merges pairwise coprime bivariate lexicographic Gröbner bases (lexGb) and an algorithm that splits a lexGb into smaller lexGbs without factorization, in a Chinese Remainder Theorem fashion.
Janet-like gröbner bases
CASC'05: Proceedings of the 8th international conference on Computer Algebra in Scientific Computing

We define a new type of Gröbner bases called Janet-like, since their properties are similar to those for Janet bases. In particular, Janet-like bases also admit an explicit formula for the Hilbert function of polynomial ideals. Cardinality of a Janet-...
Canonical comprehensive Gröbner bases
Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)

Comprehensive Gröbner bases for parametric polynomial ideals were introduced, constructed, and studied by the author in 1992. Since then the construction has been implemented in the computer algebra systems ALDES/SAC-2, MAS, REDUCE and MAPLE. A ...

Comments

Information & Contributors

Information

Published In

ISSAC '91: Proceedings of the 1991 international symposium on Symbolic and algebraic computation

June 1991

468 pages

ISBN:0897914376

DOI:10.1145/120694

Editor:
Stephen M. Watt

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: 01 June 1991

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Qualifiers

Article

Conference

ISSAC 91

Sponsor:

German Comp Soc
GMD
SIGSAM

ISSAC 91: International Symposium on Symbolic Algebraic Computation

July 15 - 17, 1991

Bonn, West Germany

Acceptance Rates

Overall Acceptance Rate 395 of 838 submissions, 47%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

9
Total Citations
View Citations
367
Total Downloads

Downloads (Last 12 months)41
Downloads (Last 6 weeks)11

Reflects downloads up to 02 Sep 2024

Other Metrics

View Author Metrics

Citations

Cited By

View all

Curtis PMaruri-Aguilar H(2016)Design for Smooth Models over Complex RegionsmODa 11 - Advances in Model-Oriented Design and Analysis10.1007/978-3-319-31266-8_9(71-78)Online publication date: 7-Jun-2016
https://doi.org/10.1007/978-3-319-31266-8_9
DERBISZ J(2014)REMARKS ON MULTIVARIATE EXTENSIONS OF POLYNOMIAL BASED SECRET SHARING SCHEMESNational Security Studies10.37055/sbn/1352386:2(285-297)Online publication date: 5-Dec-2014
https://doi.org/10.37055/sbn/135238
Montes AWibmer M(2010)Gröbner bases for polynomial systems with parametersJournal of Symbolic Computation10.1016/j.jsc.2010.06.01745:12(1391-1425)Online publication date: 1-Dec-2010
https://dl.acm.org/doi/10.1016/j.jsc.2010.06.017
Riccomagno E(2008)A short history of algebraic statisticsMetrika10.1007/s00184-008-0222-369:2-3(397-418)Online publication date: 5-Dec-2008
https://doi.org/10.1007/s00184-008-0222-3
Möller H(2005)Systems of algebraic equations solved by means of endomorphismsApplied Algebra, Algebraic Algorithms and Error-Correcting Codes10.1007/3-540-56686-4_32(43-56)Online publication date: 2-Jun-2005
https://doi.org/10.1007/3-540-56686-4_32
Grabmeier JKaltofen EWeispfenning V(2003)Cited ReferencesComputer algebra handbook10.5555/940131.940137(493-622)Online publication date: 1-Jan-2003
https://dl.acm.org/doi/10.5555/940131.940137
Weispfenning V(2003)Canonical comprehensive Gröbner basesJournal of Symbolic Computation10.1016/S0747-7171(03)00099-336:3-4(669-683)Online publication date: 1-Sep-2003
https://dl.acm.org/doi/10.1016/S0747-7171%2803%2900099-3
Weispfenning VGiusti M(2002)Canonical comprehensive Gröbner basesProceedings of the 2002 international symposium on Symbolic and algebraic computation10.1145/780506.780541(270-276)Online publication date: 10-Jul-2002
https://dl.acm.org/doi/10.1145/780506.780541
Marinari MMöller HMora T(1993)Gröbner bases of ideals defined by functionals with an application to ideals of projective pointsApplicable Algebra in Engineering, Communication and Computing10.1007/BF013868344:2(103-145)Online publication date: 1-Jun-1993
https://dl.acm.org/doi/10.1007/BF01386834

View Options

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Get Access

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

Chinese Remainder Theorem for bivariate lexicographic Gröbner bases

Janet-like gröbner bases

Canonical comprehensive Gröbner bases