Journal of Symbolic Computation最新文献

{"title":"Topological closure of formal powers series ideals and application to topological rewriting theory","authors":"Cyrille Chenavier,&nbsp;Thomas Cluzeau,&nbsp;Adya Musson-Leymarie","doi":"10.1016/j.jsc.2024.102416","DOIUrl":"10.1016/j.jsc.2024.102416","url":null,"abstract":"<div><div>We investigate formal power series ideals and their relationship to topological rewriting theory. Since commutative formal power series algebras are Zariski rings, their ideals are closed for the adic topology defined by the maximal ideal generated by the indeterminates. We provide a constructive proof of this result which, given a formal power series in the topological closure of an ideal, consists in computing a cofactor representation of the series with respect to a standard basis of the ideal. We apply this result in the context of topological rewriting theory, where two natural notions of confluence arise: topological confluence and infinitary confluence. We give explicit examples illustrating that in general, infinitary confluence is a strictly stronger notion than topological confluence. Using topological closure of ideals, we finally show that in the context of rewriting theory on commutative formal power series, infinitary and topological confluences are equivalent when the monomial order considered is compatible with the degree.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"129 ","pages":"Article 102416"},"PeriodicalIF":0.6,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143159654","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Minimal generating sets for matrix monoids","authors":"F. Hivert,&nbsp;J.D. Mitchell,&nbsp;F.L. Smith ,&nbsp;W.A. Wilson","doi":"10.1016/j.jsc.2024.102415","DOIUrl":"10.1016/j.jsc.2024.102415","url":null,"abstract":"<div><div>In this paper, we determine minimal generating sets for several well-known monoids of matrices over certain semirings. In particular, we find minimal generating sets for the monoids consisting of: all <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> boolean matrices when <span><math><mi>n</mi><mo>≤</mo><mn>8</mn></math></span>; the <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> boolean matrices containing the identity matrix (the <em>reflexive</em> boolean matrices) when <span><math><mi>n</mi><mo>≤</mo><mn>7</mn></math></span>; the <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> boolean matrices containing a permutation (the <em>Hall</em> matrices) when <span><math><mi>n</mi><mo>≤</mo><mn>8</mn></math></span>; the upper, and lower, triangular boolean matrices of every dimension; the <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> matrices over the semiring <span><math><mi>N</mi><mo>∪</mo><mo>{</mo><mo>−</mo><mo>∞</mo><mo>}</mo></math></span> with addition ⊕ defined by <span><math><mi>x</mi><mo>⊕</mo><mi>y</mi><mo>=</mo><mi>max</mi><mo>⁡</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> and multiplication ⊗ given by <span><math><mi>x</mi><mo>⊗</mo><mi>y</mi><mo>=</mo><mi>x</mi><mo>+</mo><mi>y</mi></math></span> (the <em>max-plus</em> semiring); the <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> matrices over any quotient of the max-plus semiring by the congruence generated by <span><math><mi>t</mi><mo>=</mo><mi>t</mi><mo>+</mo><mn>1</mn></math></span> where <span><math><mi>t</mi><mo>∈</mo><mi>N</mi></math></span>; the <span><math><mn>2</mn><mo>×</mo><mn>2</mn></math></span> matrices over the min-plus semiring and its finite quotients by the congruences generated by <span><math><mi>t</mi><mo>=</mo><mi>t</mi><mo>+</mo><mn>1</mn></math></span> for all <span><math><mi>t</mi><mo>∈</mo><mi>N</mi></math></span>; and the <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrices over <span><math><mi>Z</mi><mo>/</mo><mi>n</mi><mi>Z</mi></math></span> relative to their group of units.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"129 ","pages":"Article 102415"},"PeriodicalIF":0.6,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143159653","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A fast algorithm for denumerants with three variables","authors":"Feihu Liu,&nbsp;Guoce Xin","doi":"10.1016/j.jsc.2024.102414","DOIUrl":"10.1016/j.jsc.2024.102414","url":null,"abstract":"<div><div>Let <span><math><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi></math></span> be distinct positive integers such that <span><math><mi>a</mi><mo>&lt;</mo><mi>b</mi><mo>&lt;</mo><mi>c</mi></math></span> and <span><math><mi>gcd</mi><mo>⁡</mo><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. For any non-negative integer <em>n</em>, the denumerant function <span><math><mi>d</mi><mo>(</mo><mi>n</mi><mo>;</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></math></span> denotes the number of solutions of the equation <span><math><mi>a</mi><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>b</mi><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mi>c</mi><msub><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mi>n</mi></math></span> in non-negative integers <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>. We present an algorithm that computes <span><math><mi>d</mi><mo>(</mo><mi>n</mi><mo>;</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></math></span> with a time complexity of <span><math><mi>O</mi><mo>(</mo><mi>log</mi><mo>⁡</mo><mi>b</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"129 ","pages":"Article 102414"},"PeriodicalIF":0.6,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143159655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Effects of reducing redundant parameters in parameter optimization for symbolic regression using genetic programming","authors":"Gabriel Kronberger ,&nbsp;Fabrício Olivetti de França","doi":"10.1016/j.jsc.2024.102413","DOIUrl":"10.1016/j.jsc.2024.102413","url":null,"abstract":"<div><div>Gradient-based local optimization has been shown to improve results of genetic programming (GP) for symbolic regression (SR) – a machine learning method for symbolic equation learning. Correspondingly, several state-of-the-art GP implementations use iterative nonlinear least squares (NLS) algorithms for local optimization of parameters. An issue that has however mostly been ignored in SR and GP literature is overparameterization of SR expressions, and as a consequence, bad conditioning of NLS optimization problem. The aim of this study is to analyze the effects of overparameterization on the NLS results and convergence speed, whereby we use Operon as an example GP/SR implementation. First, we demonstrate that numeric rank approximation can be used to detect overparameterization using a set of six selected benchmark problems. In the second part, we analyze whether the NLS results or convergence speed can be improved by simplifying expressions to remove redundant parameters with equality saturation. This analysis is done with the much larger Feynman symbolic regression benchmark set after collecting all expressions visited by GP, as the simplification procedure is not fast enough to use it within GP fitness evaluation. We observe that Operon frequently visits overparameterized solutions but the number of redundant parameters is small on average. We analyzed the Pareto-optimal expressions of the first and last generation of GP, and found that for 70% to 80% of the simplified expressions, the success rate of reaching the optimum was better or equal than for the overparameterized form. The effect was smaller for the number of NLS iterations until convergence, where we found fewer or equal iterations for 51% to 63% of the expressions after simplification.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"129 ","pages":"Article 102413"},"PeriodicalIF":0.6,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143159656","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Limits of real bivariate rational functions","authors":"Sĩ Tiệp Đinh ,&nbsp;Feng Guo ,&nbsp;Hồng Đức Nguyễn ,&nbsp;Tiến-Sơn Phạm","doi":"10.1016/j.jsc.2024.102405","DOIUrl":"10.1016/j.jsc.2024.102405","url":null,"abstract":"<div><div>Given two nonzero polynomials <span><math><mi>f</mi><mo>,</mo><mi>g</mi><mo>∈</mo><mi>R</mi><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>]</mo></math></span> and a point <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo><mo>∈</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, we give some necessary and sufficient conditions for the existence of the limit <span><math><munder><mi>lim</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>→</mo><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></munder><mo>⁡</mo><mfrac><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mrow><mi>g</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mfrac></math></span>. We also show that, if the denominator <em>g</em> has an isolated zero at the given point <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span>, then the set of possible limits of <span><math><munder><mi>lim</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>→</mo><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></mrow></munder><mo>⁡</mo><mfrac><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mrow><mi>g</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mfrac></math></span> is a closed interval in <span><math><mover><mrow><mi>R</mi></mrow><mo>‾</mo></mover></math></span> and can be explicitly determined. As an application, we propose an effective algorithm to verify the existence of the limit and compute the limit (if it exists). Our approach is geometric and is based on Puiseux expansions.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"129 ","pages":"Article 102405"},"PeriodicalIF":0.6,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143159651","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Computing component groups of stabilizers of nilpotent orbit representatives","authors":"Emanuele Di Bella,&nbsp;Willem A. de Graaf","doi":"10.1016/j.jsc.2024.102404","DOIUrl":"10.1016/j.jsc.2024.102404","url":null,"abstract":"<div><div>We describe computational methods for computing the component group of the stabilizer of a nilpotent element in a complex simple Lie algebra. Our algorithms have been implemented in the language of the computer algebra system <span>GAP</span>4. Occasionally we need Gröbner basis computations; for these we use the systems <span>Magma</span> and <span>Singular</span>. The resulting component groups have been made available in the <span>GAP</span>4 package <span>SLA</span>.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"129 ","pages":"Article 102404"},"PeriodicalIF":0.6,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142747772","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Exact moment representation in polynomial optimization","authors":"Lorenzo Baldi,&nbsp;Bernard Mourrain","doi":"10.1016/j.jsc.2024.102403","DOIUrl":"10.1016/j.jsc.2024.102403","url":null,"abstract":"<div><div>We investigate the problem of representing moment sequences by measures in the context of Polynomial Optimization Problems, that consist in finding the infimum of a real polynomial on a real semialgebraic set defined by polynomial inequalities. We analyze the exactness of Moment Matrix (MoM) hierarchies, dual to the Sum of Squares (SoS) hierarchies, which are sequences of convex cones introduced by Lasserre to approximate measures and positive polynomials. We investigate in particular flat truncation properties, which allow testing effectively when MoM exactness holds and recovering the minimizers.</div><div>We show that the dual of the MoM hierarchy coincides with the SoS hierarchy extended with the real radical of the support of the defining quadratic module <em>Q</em>. We deduce that flat truncation happens if and only if the support of the quadratic module associated with the minimizers is of dimension zero. We also bound the order of the hierarchy at which flat truncation holds.</div><div>As corollaries, we show that flat truncation and MoM exactness hold when regularity conditions, known as Boundary Hessian Conditions, hold (and thus that MoM exactness holds generically); and when the support of the quadratic module <em>Q</em> is zero-dimensional. Effective numerical computations illustrate these flat truncation properties.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"129 ","pages":"Article 102403"},"PeriodicalIF":0.6,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143159663","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Sections of Dupin cyclides and their focal properties","authors":"Eriola Hoxhaj ,&nbsp;Jean Michel Menjanahary ,&nbsp;Rimvydas Krasauskas","doi":"10.1016/j.jsc.2024.102402","DOIUrl":"10.1016/j.jsc.2024.102402","url":null,"abstract":"<div><div>The cyclographic model of Laguerre geometry is utilized to clarify and generalize the focal properties of curves appearing as torus sections. Based on these findings, all Dupin cyclides with a given planar or spherical section are characterized as several surface families having unique extensions to certain three-orthogonal coordinate systems.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"129 ","pages":"Article 102402"},"PeriodicalIF":0.6,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143159652","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Computing the homology of universal covers via effective homology and discrete vector fields","authors":"Miguel A. Marco-Buzunáriz ,&nbsp;Ana Romero","doi":"10.1016/j.jsc.2024.102401","DOIUrl":"10.1016/j.jsc.2024.102401","url":null,"abstract":"<div><div>Effective homology techniques allow us to compute homology groups of a wide family of topological spaces. By the Whitehead tower method, this can also be used to compute higher homotopy groups. However, some of these techniques (in particular, the Whitehead tower) rely on the assumption that the starting space is simply connected. For some applications, this problem could be circumvented by replacing the space by its universal cover, which is a simply connected space that shares the higher homotopy groups of the initial space. In this paper, we formalize a simplicial construction for the universal cover, and represent it as a twisted Cartesian product.</div><div>As we show with some examples, the universal cover of a space with effective homology does not necessarily have effective homology in general. We show two independent sufficient conditions that can ensure it: one is based on a nilpotency property of the fundamental group, and the other one on discrete vector fields.</div><div>Some examples showing our implementation of these constructions in both SageMath and Kenzo are shown, together with an approach to compute the homology of the universal cover when the group is Abelian even in some cases where there is no effective homology, using the twisted homology of the space.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"128 ","pages":"Article 102401"},"PeriodicalIF":0.6,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Local dual spaces and primary decomposition","authors":"Justin Chen,&nbsp;Marc Härkönen,&nbsp;Anton Leykin","doi":"10.1016/j.jsc.2024.102400","DOIUrl":"10.1016/j.jsc.2024.102400","url":null,"abstract":"<div><div>Generalizing the concept of the Macaulay inverse system, we introduce a way to describe localizations of an ideal in a polynomial ring. This leads to an approach to the differential primary decomposition as a description of the affine scheme defined by the ideal.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"128 ","pages":"Article 102400"},"PeriodicalIF":0.6,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142651925","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}