Journal of Symbolic Computation最新文献

{"title":"Fast evaluation of generalized Todd polynomials: Applications to MacMahon's partition analysis and integer programming","authors":"Guoce Xin ,&nbsp;Yingrui Zhang ,&nbsp;ZiHao Zhang","doi":"10.1016/j.jsc.2025.102420","DOIUrl":"10.1016/j.jsc.2025.102420","url":null,"abstract":"<div><div>The Todd polynomials, denoted as <span><math><msub><mrow><mtext>td</mtext></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>m</mi></mrow></msub><mo>)</mo></math></span>, are characterized by their generating function:<span><span><span><math><munder><mo>∑</mo><mrow><mi>k</mi><mo>≥</mo><mn>0</mn></mrow></munder><msub><mrow><mtext>td</mtext></mrow><mrow><mi>k</mi></mrow></msub><msup><mrow><mi>s</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>=</mo><munderover><mo>∏</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><mfrac><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msub><mi>s</mi></mrow><mrow><msup><mrow><mi>e</mi></mrow><mrow><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msub><mi>s</mi></mrow></msup><mo>−</mo><mn>1</mn></mrow></mfrac><mo>.</mo></math></span></span></span> These polynomials serve as fundamental components in the Todd class of toric varieties – a concept of significant relevance in the study of lattice polytopes and number theory. We identify that generalized Todd polynomials emerge naturally within the realm of MacMahon's partition analysis, particularly in the context of computing the Ehrhart series. We introduce an efficient method for the evaluation of generalized Todd polynomials for numerical values of <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>. This is achieved through the development of expedited operations in the quotient ring <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>[</mo><mo>[</mo><mi>s</mi><mo>]</mo><mo>]</mo></math></span> modulo <span><math><msup><mrow><mi>s</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span>, where <em>p</em> is a large prime. The practical implications of our work are demonstrated through two applications: firstly, we facilitate a recalculated resolution of the Ehrhart series for magic squares of order 6, a problem initially addressed by the first author, reducing computation time from 70 days to approximately 1 day; secondly, we present a polynomial-time algorithm for Integer Linear Programming in the scenario where the dimension is fixed, exhibiting a notable enhancement in computational efficiency.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"130 ","pages":"Article 102420"},"PeriodicalIF":0.6,"publicationDate":"2025-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143176021","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":"Solving order 3 difference equations","authors":"Heba Bou KaedBey,&nbsp;Mark van Hoeij,&nbsp;Man Cheung Tsui","doi":"10.1016/j.jsc.2025.102419","DOIUrl":"10.1016/j.jsc.2025.102419","url":null,"abstract":"<div><div>We classify order 3 linear difference operators over <span><math><mi>C</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> that are solvable in terms of lower order difference operators. To prove this result, we introduce the notion of absolute irreducibility for difference modules, and classify modules of arbitrary dimension that are irreducible but not absolutely irreducible.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"129 ","pages":"Article 102419"},"PeriodicalIF":0.6,"publicationDate":"2025-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143159664","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":"Computational Algebra and Geometry: A special issue in memory and honor of Agnes Szanto","authors":"Carlos D'Andrea,&nbsp;Hoon Hong,&nbsp;Evelyne Hubert,&nbsp;Teresa Krick","doi":"10.1016/j.jsc.2024.102418","DOIUrl":"10.1016/j.jsc.2024.102418","url":null,"abstract":"","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"129 ","pages":"Article 102418"},"PeriodicalIF":0.6,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143160245","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":"Quadratic equations in the lamplighter group","authors":"Alexander Ushakov,&nbsp;Chloe Weiers","doi":"10.1016/j.jsc.2024.102417","DOIUrl":"10.1016/j.jsc.2024.102417","url":null,"abstract":"<div><div>In this paper we study the complexity of solving quadratic equations in the lamplighter group. We give a complete classification of cases (depending on genus and other characteristics of a given equation) when the problem is <strong>NP</strong>-complete or polynomial-time decidable. We notice that the conjugacy problem can be solved in linear time. Finally, we prove that the problem belongs to the class <strong>XP</strong>.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"129 ","pages":"Article 102417"},"PeriodicalIF":0.6,"publicationDate":"2024-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143128507","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":"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}