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Tail reduction free term rewriting systems revisited 重新访问了尾约简自由项重写系统
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-06-11 DOI: 10.1016/j.jsc.2025.102474
Sándor Vágvölgyi
{"title":"Tail reduction free term rewriting systems revisited","authors":"Sándor Vágvölgyi","doi":"10.1016/j.jsc.2025.102474","DOIUrl":"10.1016/j.jsc.2025.102474","url":null,"abstract":"<div><div>First we present various undecidability results on numerous subclasses of tail reduction free term rewriting systems which simply follow from the literature review on term rewriting. Then we show that the following problems are undecidable for linear tail reduction free term rewriting systems: the word problem, the existence of normal forms problem, the common ancestor problem, the joinability problem, the normalizing problem, the termination problem, the convergence problem, the reflexive transitive closure of reduction relation inclusion problem, the reflexive transitive closure of reduction relation equality problem, and the reflexive transitive closure of reduction relation proper inclusion problem. Finally, we show that the following problems are undecidable for right-linear trf TRSs: the inductive problem, the congruence relation inclusion problem, the congruence relation equality problem, and the congruence relation proper inclusion problem. In addition, we show that the restrictions of all the problems to ground terms are also undecidable.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102474"},"PeriodicalIF":0.6,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144270009","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}
引用次数: 0
Testing containment of tropical hypersurfaces within polynomial complexity 多项式复杂度下热带超曲面的包容性检验
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-06-11 DOI: 10.1016/j.jsc.2025.102472
Dima Grigoriev
{"title":"Testing containment of tropical hypersurfaces within polynomial complexity","authors":"Dima Grigoriev","doi":"10.1016/j.jsc.2025.102472","DOIUrl":"10.1016/j.jsc.2025.102472","url":null,"abstract":"<div><div>For tropical <em>n</em>-variable polynomials <span><math><mi>f</mi><mo>,</mo><mi>g</mi></math></span> a criterion of containment for tropical hypersurfaces <span><math><mrow><mi>Trop</mi></mrow><mo>(</mo><mi>f</mi><mo>)</mo><mo>⊂</mo><mrow><mi>Trop</mi></mrow><mo>(</mo><mi>g</mi><mo>)</mo></math></span> is provided in terms of their Newton polyhedra <span><math><mi>N</mi><mo>(</mo><mi>f</mi><mo>)</mo><mo>,</mo><mi>N</mi><mo>(</mo><mi>g</mi><mo>)</mo><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></math></span>. Namely, <span><math><mrow><mi>Trop</mi></mrow><mo>(</mo><mi>f</mi><mo>)</mo><mo>⊂</mo><mrow><mi>Trop</mi></mrow><mo>(</mo><mi>g</mi><mo>)</mo></math></span> iff for every vertex <em>v</em> of <span><math><mi>N</mi><mo>(</mo><mi>g</mi><mo>)</mo></math></span> there exists a unique vertex <em>w</em> of <span><math><mi>N</mi><mo>(</mo><mi>f</mi><mo>)</mo></math></span> such that for the tangent cones it holds <span><math><mi>v</mi><mo>−</mo><mi>w</mi><mo>+</mo><mi>N</mi><msub><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mrow><mi>w</mi></mrow></msub><mo>⊆</mo><mi>N</mi><msub><mrow><mo>(</mo><mi>g</mi><mo>)</mo></mrow><mrow><mi>v</mi></mrow></msub></math></span>. Relying on this criterion an algorithm is designed which tests whether <span><math><mrow><mi>Trop</mi></mrow><mo>(</mo><mi>f</mi><mo>)</mo><mo>⊂</mo><mrow><mi>Trop</mi></mrow><mo>(</mo><mi>g</mi><mo>)</mo></math></span> within polynomial complexity.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102472"},"PeriodicalIF":0.6,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144270130","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}
引用次数: 0
How to generate all possible rational Wilf–Zeilberger forms? 如何生成所有可能的理性Wilf-Zeilberger形式？
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-06-11 DOI: 10.1016/j.jsc.2025.102473
Shaoshi Chen , Christoph Koutschan , Yisen Wang
{"title":"How to generate all possible rational Wilf–Zeilberger forms?","authors":"Shaoshi Chen ,&nbsp;Christoph Koutschan ,&nbsp;Yisen Wang","doi":"10.1016/j.jsc.2025.102473","DOIUrl":"10.1016/j.jsc.2025.102473","url":null,"abstract":"<div><div>Wilf–Zeilberger pairs are fundamental in the algorithmic theory of Wilf and Zeilberger for computer-generated proofs of combinatorial identities. Wilf–Zeilberger forms are their high-dimensional generalizations, which can be used for proving and discovering convergence acceleration formulas. This paper presents a structural description of all possible rational such forms, which can be viewed as an additive analog of the classical Ore–Sato theorem. Based on this analog, we show a structural decomposition of so-called multivariate hyperarithmetic expressions, which extend multivariate hypergeometric terms to the additive setting.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102473"},"PeriodicalIF":0.6,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144322029","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}
引用次数: 0
Constructively describing orbit spaces of finite groups by few inequalities 用少量不等式构造描述有限群的轨道空间
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-05-30 DOI: 10.1016/j.jsc.2025.102471
Philippe Moustrou , Cordian Riener , Robin Schabert
{"title":"Constructively describing orbit spaces of finite groups by few inequalities","authors":"Philippe Moustrou ,&nbsp;Cordian Riener ,&nbsp;Robin Schabert","doi":"10.1016/j.jsc.2025.102471","DOIUrl":"10.1016/j.jsc.2025.102471","url":null,"abstract":"<div><div>Let <em>G</em> be a finite group acting linearly on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. A celebrated Theorem of Procesi and Schwarz gives an explicit description of the orbit space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>/</mo><mo>/</mo><mi>G</mi></math></span> as a basic closed semi-algebraic set. We give a new proof of this statement and another description as a basic closed semi-algebraic set using elementary tools from real algebraic geometry. Bröcker was able to show that the number of inequalities needed to describe the orbit space generically depends only on the group <em>G</em>. Here, we construct such inequalities explicitly for abelian groups and in the case where only one inequality is needed. Furthermore, we answer an open question raised by Bröcker concerning the genericity of his result.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102471"},"PeriodicalIF":0.6,"publicationDate":"2025-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144223225","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}
引用次数: 0
Geometric complexity theory for product-plus-power 乘积加幂的几何复杂性理论
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-05-28 DOI: 10.1016/j.jsc.2025.102458
Pranjal Dutta , Fulvio Gesmundo , Christian Ikenmeyer , Gorav Jindal , Vladimir Lysikov
{"title":"Geometric complexity theory for product-plus-power","authors":"Pranjal Dutta ,&nbsp;Fulvio Gesmundo ,&nbsp;Christian Ikenmeyer ,&nbsp;Gorav Jindal ,&nbsp;Vladimir Lysikov","doi":"10.1016/j.jsc.2025.102458","DOIUrl":"10.1016/j.jsc.2025.102458","url":null,"abstract":"<div><div>According to Kumar's recent surprising result (ToCT'20), a small border Waring rank implies that the polynomial can be approximated as a sum of a constant and a small product of linear polynomials. We prove the converse of Kumar's result and establish a tight connection between border Waring rank and the model of computation in Kumar's result. In this way, we obtain a new formulation of border Waring rank, up to a factor of the degree.</div><div>We connect this new formulation to the orbit closure problem of the product-plus-power polynomial. We study this orbit closure from two directions:</div><div>1. We deborder this orbit closure and some related orbit closures, i.e., prove all points in the orbit closure have small non-border algebraic branching programs.</div><div>2. We fully implement the geometric complexity theory approach against the power sum by generalizing the ideas of Ikenmeyer-Kandasamy (STOC'20) to this new orbit closure. In this way, we obtain new multiplicity obstructions that are constructed from just the symmetries of the polynomials.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102458"},"PeriodicalIF":0.6,"publicationDate":"2025-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144223224","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}
引用次数: 0
Viro's patchworking and the signed reduced A-discriminant 维罗的拼接和带符号的减a辨别式
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-05-26 DOI: 10.1016/j.jsc.2025.102462
Weixun Deng , J. Maurice Rojas , Máté L. Telek
{"title":"Viro's patchworking and the signed reduced A-discriminant","authors":"Weixun Deng ,&nbsp;J. Maurice Rojas ,&nbsp;Máté L. Telek","doi":"10.1016/j.jsc.2025.102462","DOIUrl":"10.1016/j.jsc.2025.102462","url":null,"abstract":"<div><div>Computing the isotopy type of a hypersurface, defined as the positive real zero set of a multivariate polynomial, is a challenging problem in real algebraic geometry. We focus on the case where the defining polynomial has combinatorially restricted exponent vectors and fixed coefficient signs, enabling faster computation of the isotopy type. In particular, Viro's patchworking provides a polyhedral complex that has the same isotopy type as the hypersurface, for certain choices of the coefficients. So we present properties of the signed support, focusing mainly on the case of n-variate (n+3)-nomials, that ensure all possible isotopy types can be obtained via patchworking. To prove this, we study the signed reduced A-discriminant and show that it has a simple structure if the signed support satisfies some combinatorial conditions.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102462"},"PeriodicalIF":0.6,"publicationDate":"2025-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144167100","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}
引用次数: 0
Positivity proofs for linear recurrences through contracted cones 缩锥线性递推的正性证明
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-05-22 DOI: 10.1016/j.jsc.2025.102463
Alaa Ibrahim, Bruno Salvy
{"title":"Positivity proofs for linear recurrences through contracted cones","authors":"Alaa Ibrahim,&nbsp;Bruno Salvy","doi":"10.1016/j.jsc.2025.102463","DOIUrl":"10.1016/j.jsc.2025.102463","url":null,"abstract":"<div><div>Deciding the positivity of a sequence defined by a linear recurrence with polynomial coefficients and initial condition is difficult in general. Even in the case of recurrences with constant coefficients, it is known to be decidable only for order up to 5. We consider a large class of linear recurrences of arbitrary order, with polynomial coefficients, for which an algorithm decides positivity for initial conditions outside of a hyperplane. The underlying algorithm constructs a cone, contracted by the recurrence operator, that allows a proof of positivity by induction. The existence and construction of such cones relies on the extension of the classical Perron-Frobenius theory to matrices leaving a cone invariant.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102463"},"PeriodicalIF":0.6,"publicationDate":"2025-05-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144147708","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}
引用次数: 0
Computing implicitizations of multi-graded polynomial maps 多阶多项式映射的计算隐式
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-05-20 DOI: 10.1016/j.jsc.2025.102459
Joseph Cummings , Benjamin Hollering
{"title":"Computing implicitizations of multi-graded polynomial maps","authors":"Joseph Cummings ,&nbsp;Benjamin Hollering","doi":"10.1016/j.jsc.2025.102459","DOIUrl":"10.1016/j.jsc.2025.102459","url":null,"abstract":"<div><div>In this paper, we focus on computing the kernel of a map of polynomial rings. This core problem in symbolic computation is known as implicitization. While Gröbner basis methods can be used to solve this problem, these methods can become infeasible as the number of variables increases. In the case when the polynomial map is multigraded, we consider an alternative approach. We first demonstrate how to quickly compute a matrix of maximal rank for which a polynomial map has a positive multigrading. We then describe how minimal generators in each graded component of the kernel can be computed with linear algebra. We have implemented our techniques in Macaulay2 and show that our implementation can compute many generators of low degree in examples where standard techniques have failed. This includes several examples coming from phylogenetics where even a complete list of quadrics and cubics were unknown. When the multigrading refines total degree, our algorithm is <em>embarassingly parallel</em>. A fully parallelized version of our algorithm is in development in both Macaulay2 and OSCAR.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102459"},"PeriodicalIF":0.6,"publicationDate":"2025-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144134496","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}
引用次数: 0
The convex algebraic geometry of higher-rank numerical ranges 高阶数值范围的凸代数几何
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-05-19 DOI: 10.1016/j.jsc.2025.102457
Jonathan Niño-Cortés, Cynthia Vinzant
{"title":"The convex algebraic geometry of higher-rank numerical ranges","authors":"Jonathan Niño-Cortés,&nbsp;Cynthia Vinzant","doi":"10.1016/j.jsc.2025.102457","DOIUrl":"10.1016/j.jsc.2025.102457","url":null,"abstract":"<div><div>The higher-rank numerical range is a convex compact set generalizing the classical numerical range of a square complex matrix, first appearing in the study of quantum error correction. We will discuss some of the real algebraic and convex geometry of these sets, including a generalization of Kippenhahn's theorem, and describe an algorithm to explicitly calculate the higher-rank numerical range of a given matrix.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102457"},"PeriodicalIF":0.6,"publicationDate":"2025-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144147707","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}
引用次数: 0
Extremal decompositions of tropical varieties and relations with rigidity theory 热带品种的极值分解及其与刚性理论的关系
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2025-05-19 DOI: 10.1016/j.jsc.2025.102461
Farhad Babaee, Sean Dewar, James Maxwell
{"title":"Extremal decompositions of tropical varieties and relations with rigidity theory","authors":"Farhad Babaee,&nbsp;Sean Dewar,&nbsp;James Maxwell","doi":"10.1016/j.jsc.2025.102461","DOIUrl":"10.1016/j.jsc.2025.102461","url":null,"abstract":"<div><div>Extremality and irreducibility constitute fundamental concepts in mathematics, particularly within tropical geometry. While extremal decomposition is typically computationally hard, this article presents a fast algorithm for identifying the extremal decomposition of tropical varieties with rational balanced weightings. Additionally, we explore connections and applications related to rigidity theory. In particular, we prove that a tropical hypersurface is extremal if and only if it has a unique reciprocal diagram up to homothety. We further show that our approach also allows for computing Chow Betti numbers for complete toric varieties.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102461"},"PeriodicalIF":0.6,"publicationDate":"2025-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144125166","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}
引用次数: 0
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