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Buchberger's algorithm: The term rewriter's point of view

Reinhard Bündgen
<span title="">1996</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/elaf5sq7lfdxfdejhkqbtz6qoq" style="color: black;">Theoretical Computer Science</a> </i> &nbsp;
The division operation needed in Buchberger's algorithm reduces to a narrowing procedure which becomes part of the critical pair computation process.  ...  We analyse the relations between completion procedures for polynomials and terms and thereby show how Buchberger's algorithm for multivariate polynomials over finite fields and over the rationals can be  ...  We do not propose to replace Buchberger's algorithm by a term completion procedure. On the contrary, Buchberger's algorithm is in general much more efficient than its simulations by term completion.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0304-3975(95)00061-5">doi:10.1016/0304-3975(95)00061-5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7htfv4v4ebhd3j2fetbhhqjf4u">fatcat:7htfv4v4ebhd3j2fetbhhqjf4u</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170923231138/http://publisher-connector.core.ac.uk/resourcesync/data/elsevier/pdf/094/aHR0cDovL2FwaS5lbHNldmllci5jb20vY29udGVudC9hcnRpY2xlL3BpaS8wMzA0Mzk3NTk1MDAwNjE1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/70/7a/707a2fcefb779ec3d2c3ae17a645a08c77509254.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/0304-3975(95)00061-5"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> elsevier.com </button> </a>

Continuously Parameterized Symmetries and Buchberger's Algorithm

Ralf Hemmecke
<span title="">2002</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ezljl2d3lzga5efenbxdvvfcpa" style="color: black;">Journal of symbolic computation</a> </i> &nbsp;
In solving such a system using Buchberger's algorithm, the symmetries are neglected.  ...  This paper presents a method that shows how this can be achieved and also gives an algorithm that brings together continuously parameterized symmetries with Buchberger's algorithm.  ...  bases B i and corresponding "constraints" C i such that Z(B, C) = i Z(B i , C i ) holds.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1006/jsco.2001.0478">doi:10.1006/jsco.2001.0478</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/d4c4nwbb2vcjfm6dgsoykr6ezy">fatcat:d4c4nwbb2vcjfm6dgsoykr6ezy</a> </span>
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Buchberger's algorithm: A constraint-based completion procedure [chapter]

Leo Bachmair, Harald Ganzinger
<i title="Springer-Verlag"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/aoiblffcszb7tnspm6ro36gbtm" style="color: black;">Constraints in Computational Logics</a> </i> &nbsp;
We present an extended completion procedure with builtin theories de ned by a collection of associativity and commutativity axioms and additional ground equations, and reformulate Buchberger's algorithm  ...  The presentation of completion is at an abstract level, by transition rules, with a suitable notion of fairness used to characterize a wide class of correct completion procedures, among them Buchberger's  ...  We represent Buchberger's algorithm as an (extended) completion procedure with a built-in theory AC B.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/bfb0016860">doi:10.1007/bfb0016860</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/ccl/BachmairG94.html">dblp:conf/ccl/BachmairG94</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/v5nmok4co5c65jdrrdsdckpiha">fatcat:v5nmok4co5c65jdrrdsdckpiha</a> </span>
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Gröbner Bases of Modules and Faugère's F_4 Algorithm in Isabelle/HOL [article]

Alexander Maletzky, Fabian Immler
<span title="2018-05-01">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The formalization covers all of the essentials of the theory (polynomial reduction, S-polynomials, Buchberger's algorithm, Buchberger's criteria for avoiding useless pairs), but also includes more advanced  ...  All formalized algorithms can be translated into executable code operating on concrete data structures, enabling the certified computation of (reduced) Gröbner bases and syzygy modules.  ...  This procedure is called Buchberger's algorithm, which is justified by Theorem 5 (Buchberger, 1965) .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1805.00304v1">arXiv:1805.00304v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/rnx5vz5c3zhl7lxqyyvy6iy6je">fatcat:rnx5vz5c3zhl7lxqyyvy6iy6je</a> </span>
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Reinforcement Learning in Buchberger's Algorithm

Dylan Peifer
<span title="">2021</span>
In this thesis we introduce a new approach to Buchberger's algorithm that uses deep reinforcement learning agents to perform S-pair selection, a key choice in the algorithm.  ...  s algorithm is the classical algorithm for computing a Gröbner basis, and optimized implementations are crucial for many computer algebra systems.  ...  Chapter GRÖBNER BASES This chapter is a friendly introduction to Gröbner bases and Buchberger's algorithm.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.7298/4fpv-bn09">doi:10.7298/4fpv-bn09</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fonlips6izcrvi6ewbcbgsdy6q">fatcat:fonlips6izcrvi6ewbcbgsdy6q</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20210915121958/https://ecommons.cornell.edu/bitstream/handle/1813/109782/Peifer_cornellgrad_0058F_12500.pdf;jsessionid=390C94E625C700AB0EE618C789DCE9EB?sequence=1" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/e1/f2/e1f268c97080e01222d17b5a03ee5b6003baef0e.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.7298/4fpv-bn09"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Gröbner Bases of Modules and Faugère's $$F_4$$F4 Algorithm in Isabelle/HOL [chapter]

Alexander Maletzky, Fabian Immler
<span title="">2018</span> <i title="Springer International Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
The formalization covers all of the essentials of the theory (polynomial reduction, S-polynomials, Buchberger's algorithm, Buchberger's criteria for avoiding useless pairs), but also includes more advanced  ...  All formalized algorithms can be translated into executable code operating on concrete data structures, enabling the certified computation of (reduced) Gröbner bases and syzygy modules.  ...  This procedure is called Buchberger's algorithm, which is justified by Theorem 5 (Buchberger, 1965) .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-319-96812-4_16">doi:10.1007/978-3-319-96812-4_16</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/g3idhjf4grdthnq2rmoiynf7ta">fatcat:g3idhjf4grdthnq2rmoiynf7ta</a> </span>
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Signature-based Möller's algorithm for strong Gröbner bases over PIDs [article]

Maria Francis, Thibaut Verron
<span title="2019-01-28">2019</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this paper, we present a signature-based version of M\"oller's classical variant of Buchberger's algorithm for computing strong Gr\"obner bases over Principal Ideal Domains (or PIDs).  ...  Signature-based algorithms are the latest and most efficient approach as of today to compute Gr\"obner bases for polynomial systems over fields.  ...  Mora for a fruitful discussion on the syzygy paradigm for Gröbner basis algorithms, and M. Kauers for his valuable insights and comments all through the elaboration of this work.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1901.09586v1">arXiv:1901.09586v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/pyswlsrpsrh3fgu5vudxs5nnhq">fatcat:pyswlsrpsrh3fgu5vudxs5nnhq</a> </span>
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A New Proof for the Correctness of F5 (F5-Like) Algorithm [article]

Yao Sun, Dingkang Wang
<span title="2010-06-30">2010</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this paper, the F5B algorithm is briefly revisited and a new complete proof for the correctness of F5B algorithm is proposed.  ...  A natural variation of F5 algorithm, which transforms original F5 algorithm to a non-incremental algorithm, is presented and proved in this paper as well.  ...  Conclusion In this paper, a complete proof for the correctness of F5 (F5-like) algorithm is presented.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1004.0084v4">arXiv:1004.0084v4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/stpjipy6ejg3bchcehtkt4qr3a">fatcat:stpjipy6ejg3bchcehtkt4qr3a</a> </span>
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An algorithm for the construction of matrix representationsfor finitely presented non-commutative algebras

Gilles Labonté
<span title="">1990</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/ezljl2d3lzga5efenbxdvvfcpa" style="color: black;">Journal of symbolic computation</a> </i> &nbsp;
A set of vector constraints which is given as part of the initial data determines which particular representation of E is produced.  ...  We study an algorithm for the construction, from this presentation, of linear, i.e, matrix, representations of this algebra.  ...  This is the same result as obtained through the use of Buchberger's algorithm. 6.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1016/s0747-7171(08)80004-1">doi:10.1016/s0747-7171(08)80004-1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/73aexvqwy5d6japwgpjzzojgpa">fatcat:73aexvqwy5d6japwgpjzzojgpa</a> </span>
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An Hilbert-Driven Strategy for Signature-Based Gröbner Basis Algorithms [chapter]

Bruno Simões
<span title="">2014</span> <i title="Springer London"> Future Vision and Trends on Shapes, Geometry and Algebra </i> &nbsp;
We present a new criterion to improve the performance of Gröbner bases algorithms, driven by polynomial signatures.  ...  One of the major limitations of the algorithms to compute Gröbner bases is their computation time.  ...  Earlier approaches to solve large constraint systems of this type have relied on semi-automatic procedures but automated approaches based on problem modeling have emerged recently in the literature, thereby  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-1-4471-6461-6_2">doi:10.1007/978-1-4471-6461-6_2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ce5oemdrn5eopnznhq37y2qlmi">fatcat:ce5oemdrn5eopnznhq37y2qlmi</a> </span>
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Page 4508 of Mathematical Reviews Vol. , Issue 97G [page]

<span title="">1997</span> <i title="American Mathematical Society"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_mathematical-reviews" style="color: black;">Mathematical Reviews </a> </i> &nbsp;
{For the entire collection see MR 97e:68003. } 97g:68105 68Q40 Bachmair, Leo (1-SUNYS-C; Stony Brook, NY); Ganzinger, Harald (D-MPI-I; Saarbriicken) Buchberger’s algorithm: a constraint-based completion  ...  SCIENCE 4508 late Buchberger’s algorithm for constructing Grébner bases for polynomial ideas in this formalism.  ... 
<span class="external-identifiers"> </span>
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Extended parallelism in the Gr�bner basis algorithm

Stephen A. Schwab
<span title="">1992</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/6ni4hdyv7zdtzjxnzscp53l5ry" style="color: black;">International journal of parallel programming</a> </i> &nbsp;
This paper presents a new parallel implementation to compute Gr6bner bases utilizing two different forms of parallelism.  ...  A hybrid algorithm that outperforms both of the original approaches is presented.  ...  The interested reader is referred to Buchberger's work, (2"3) where one will find a complete introduction to Gr6bner bases, a description of their computation, as well as a list of many applications.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/bf01379314">doi:10.1007/bf01379314</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/fvloeanwijerpfru5arfrogj5i">fatcat:fvloeanwijerpfru5arfrogj5i</a> </span>
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Algorithms for solving reachability problems in 2-link planar arms using Gröbner bases

Zahra Nilforoushan, Keivan Borna
<span title="">2014</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/u7ltb3rqybdztn2wjutosgiekq" style="color: black;">Mathematical Sciences</a> </i> &nbsp;
Furthermore, we proposed some algorithms for a special case where the shoulder of an arm moves along a given segment or passes through a certain path.  ...  It is essential to mention that our approach is based on the Gröbner bases technique.  ...  In the course of computing Gröbner bases by Buchberger's algorithm, one has to compute and reduce numerous so called S-polynomials.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s40096-015-0139-8">doi:10.1007/s40096-015-0139-8</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ecgdolg7izcubkfxshtqy3sbm4">fatcat:ecgdolg7izcubkfxshtqy3sbm4</a> </span>
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On Gröbner Bases in the Context of Satisfiability-Modulo-Theories Solving over the Real Numbers [chapter]

Sebastian Junges, Ulrich Loup, Florian Corzilius, Erika Ábrahám
<span title="">2013</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
We modify Buchberger's algorithm by implementing a new update operator to optimize the Gröbner basis and provide two methods to handle inequalities.  ...  SMT solvers combine a SAT solver to resolve the Boolean structure of a given formula with theory solvers to verify the consistency of sets of theory constraints.  ...  In Section 4.1 we discuss how its design wraps a GB procedure such as Buchberger's algorithm, while leaving the GB procedure itself untouched.  ... 
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<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170922213143/http://sunsite.informatik.rwth-aachen.de/Publications/AIB/2013/2013-08.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/98/ba/98ba9eac332f6e2cbe2962f4f83072f1a7a08347.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-642-40663-8_18"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Polynomial Equivalence Problems: Algorithmic and Theoretical Aspects [chapter]

Jean-Charles Faugère, Ludovic Perret
<span title="">2006</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
Concerning the practical aspect, we present a new algorithm for solving IP.  ...  From a practical point of view, we employed a fast Gröbner basis algorithm, namely F5 [17], for solving this system.  ...  The historical method for computing Gröbner bases is Buchberger's algorithm [9, 8] . Recently, more efficient algorithms have been proposed.  ... 
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<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170812043534/https://www.iacr.org/cryptodb/archive/2006/EUROCRYPT/2347/2347.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/d2/ae/d2aefdccff388b5d384d8974a0acc1163ba57d57.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/11761679_3"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>
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