Towards signature-based gröbner basis algorithms for computing the nondegenerate locus of a polynomial system
Pages 41 - 45
Abstract
Problem statement. Let K be a field and K be an algebraic closure of K. Consider the polynomial ring R = K[x1,..., xn] over K and a finite sequence of polynomials f1,...,fc in R with c ≤ n. Let V ⊂ Kn be the algebraic set defined by the simultaneous vanishing of the fi's. Recall that V can be decomposed into finitely many irreducible components, whose codimension cannot be greater than c. The set Vc which is the union of all these irreducible components of codimension exactly c is named further the nondegenerate locus of f1,...,fc.
References
[1]
Philippe Aubry, Daniel Lazard, and Marc Moreno Maza. On the Theories of Triangular Sets. 28(1):105--124.
[2]
Thomas Becker and Volker Weispfenning. Gröbner bases, volume 141 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1993. A computational approach to commutative algebra, In cooperation with Heinz Kredel.
[3]
Jeff Bezanson, Alan Edelman, Stefan Karpinski, and Viral B Shah. Julia: A fresh approach to numerical computing. 59(1):65--98.
[4]
Wieb Bosma, John Cannon, and Catherine Playoust. The Magma algebra system. I. The user language. 24(3--4):235--265.
[5]
Massimo Caboara, Pasqualina Conti, and Carlo Traverse. Yet another ideal decomposition algorithm. In Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, Lecture Notes in Computer Science, pages 39--54. Springer.
[6]
Shang-Ching Chou and Xiao-Shan Gao. Ritt-Wu's decomposition algorithm and geometry theorem proving. In 10th International Conference on Automated Deduction, Lecture Notes in Computer Science, pages 207--220. Springer.
[7]
Wolfram Decker, Gert-Martin Greuel, and Gerhard Pfister. Primary Decomposition: Algorithms and Comparisons. In Algorithmic Algebra and Number Theory, pages 187--220. Springer.
[8]
Wolfram Decker, Gert-Martin Greuel, Gerhard Pfister, and Hans Schönemann. Singular 4-3-0 --- A computer algebra system for polynomial computations.
[9]
Christian Eder and Jean-Charles Faugère. A survey on signature-based algorithms for computing gröbner bases. 80:719--784.
[10]
David Eisenbud, Craig Huneke, and Wolmer Vasconcelos. Direct methods for primary decomposition. 110(1):207--235.
[11]
Jean-Charles Faugère. A new efficient algorithm for computing gröbner bases without reduction to zero (f5). In ISSAC'02, pages 75--83.
[12]
Shuhong Gao, Yinhua Guan, and Frank Volny IV. A new incremental algorithm for computing gröbner bases. In Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, pages 13--19, 2010.
[13]
Patrizia Gianni, Barry Trager, and Gail Zacharias. Gröbner bases and primary decomposition of polynomial ideals. 6(2):149--167.
[14]
Daniel R. Grayson and Michael E. Stillman. Macaulay2, a software system for research in algebraic geometry. Available at http://www.math.uiuc.edu/Macaulay2/.
[15]
Gert-Martin Greuel and Gerhard Pfister. A Singular Introduction to Commutative Algebra. Springer Berlin Heidelberg, 2 edition.
[16]
Evelyne Hubert. Notes on Triangular Sets and Triangulation-Decomposition Algorithms I. In Symbolic and Numerical Scientific Computation, Lecture Notes in Computer Science, pages 1--39. Springer.
[17]
Teresa Krick and Alessandro Logar. An algorithm for the computation of the radical of an ideal in the ring of polynomials. In AAECC 1991, pages 195--205. Springer-Verlag.
[18]
Grégoire Lecerf. Computing an equidimensional decomposition of an algebraic variety by means of geometric resolutions. In ISSAC'00, pages 209--216.
[19]
Grégoire Lecerf. Computing the equidimensional decomposition of an algebraic closed set by means of lifting fibers. 19(4):564--596.
[20]
François Lemaire, Marc Moreno Maza, Wei Pan, and Yuzhen Xie. When does ⟨t⟩ equal sat(t)? 46(12):1291--1305.
[21]
Dongming Wang. An Elimination Method for Polynomial Systems. 16(2):83--114.
[22]
Dongming Wang. Elimination Methods. Texts and Monographs in Symbolic Computation. Springer Vienna.
Recommendations
Towards Mixed Gröbner Basis Algorithms: the Multihomogeneous and Sparse Case
ISSAC '18: Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic ComputationOne of the biggest open problems in computational algebra is the design of efficient algorithms for Gröbner basis computations that take into account the sparsity of the input polynomials. We can perform such computations in the case of unmixed ...
Computing a Gröbner basis of a polynomial ideal over a Euclidean domain
An algorithm for computing a Grobner basis of a polynomial ideal over a Euclidean domain is presented. The algorithm takes an ideal specified by a finite set of polynomials as its input; it produces another finite basis of the same ideal with the ...
A signature-based algorithm for computing the nondegenerate locus of a polynomial system
AbstractPolynomial system solving arises in many application areas to model non-linear geometric properties. In such settings, polynomial systems may come with degeneration which the end-user wants to exclude from the solution set. The nondegenerate ...
Comments
Please enable JavaScript to view thecomments powered by Disqus.Information & Contributors
Information
Published In
Copyright © 2022 Copyright is held by the owner/author(s).
Permission to make digital or hard copies of part or all 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 third-party components of this work must be honored. For all other uses, contact the Owner/Author.
Publisher
Association for Computing Machinery
New York, NY, United States
Publication History
Published: 23 November 2022
Published in SIGSAM-CCA Volume 56, Issue 2
Check for updates
Qualifiers
- Research-article
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 31Total Downloads
- Downloads (Last 12 months)6
- Downloads (Last 6 weeks)1
Reflects downloads up to 20 Nov 2024
Other Metrics
Citations
View Options
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in