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Supercongruences involving products of two binomial coefficients modulo p4 涉及两个二项式系数的乘积模p4的超同余
IF 1.1 4区 数学
Journal of Symbolic Computation Pub Date : 2026-07-01 Epub Date: 2025-12-22 DOI: 10.1016/j.jsc.2025.102545
Guo-Shuai Mao
{"title":"Supercongruences involving products of two binomial coefficients modulo p4","authors":"Guo-Shuai Mao","doi":"10.1016/j.jsc.2025.102545","DOIUrl":"10.1016/j.jsc.2025.102545","url":null,"abstract":"<div><div>In this paper, we mainly prove a challenging congruence conjecture of Z.-W. Sun (<span><span>2014</span></span>) via symbolic summation: Let <span><math><mi>p</mi><mo>&gt;</mo><mn>5</mn></math></span> be a prime. Then<span><span><span><math><munderover><mo>∑</mo><mrow><mi>k</mi><mo>=</mo><mo>(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mn>2</mn></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></munderover><mfrac><mrow><msup><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mn>2</mn><mi>k</mi></mrow></mtd></mtr><mtr><mtd><mi>k</mi></mtd></mtr></mtable><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mi>k</mi><msup><mrow><mn>16</mn></mrow><mrow><mi>k</mi></mrow></msup></mrow></mfrac><mo>≡</mo><mo>−</mo><mfrac><mrow><mn>21</mn></mrow><mrow><mn>2</mn></mrow></mfrac><msub><mrow><mi>H</mi></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msub><mspace></mspace><mo>(</mo><mrow><mi>mod</mi></mrow><mspace></mspace><msup><mrow><mi>p</mi></mrow><mrow><mn>4</mn></mrow></msup><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> denotes the <em>n</em>-th harmonic number. The necessary proofs are provided by the computer algebra software Sigma to find and verify the underlying hypergeometric sum identities.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102545"},"PeriodicalIF":1.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145839832","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
Integer programming with binary and bounded variables via Gröbner bases with applications to multiobjective integer programming 基于Gröbner的二进制和有界变量的整数规划及其在多目标整数规划中的应用
IF 1.1 4区 数学
Journal of Symbolic Computation Pub Date : 2026-07-01 Epub Date: 2025-11-19 DOI: 10.1016/j.jsc.2025.102529
Gabriel Mattos Langeloh
{"title":"Integer programming with binary and bounded variables via Gröbner bases with applications to multiobjective integer programming","authors":"Gabriel Mattos Langeloh","doi":"10.1016/j.jsc.2025.102529","DOIUrl":"10.1016/j.jsc.2025.102529","url":null,"abstract":"<div><div>Families of integer programming problems can be solved efficiently in practice once their reduced Gröbner basis is known. However, computing Gröbner bases is often hard, especially when binary or integer bounded variables are present in the problem formulation. In this paper, we study the specific structure of the constraint matrix of integer programs with bounded variables and the implications of this structure to the truncated Gröbner bases of integer programming problems. In this direction, we introduce a new Binary Truncation Criterion that is capable of predicting and eliminating useless S-vectors before they built in the Gröbner basis computation. Additionally, we propose improvements to the Gröbner basis approach to multiobjective integer programming of <span><span>Jiménez-Tafur (2017)</span></span> and <span><span>Hartillo-Hermoso et al. (2020)</span></span>, such as a proof that truncated Gröbner bases can be used in their algorithm with no loss of correctness, implying that our new truncation techniques are also useful in this application. All new proposed methods are implemented in the open source package IPGBs and their performance is empirically validated.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102529"},"PeriodicalIF":1.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145580128","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
Wronski pairs of honeycomb curves 蜂窝曲线的朗斯基对
IF 1.1 4区 数学
Journal of Symbolic Computation Pub Date : 2026-07-01 Epub Date: 2025-11-17 DOI: 10.1016/j.jsc.2025.102528
Laura Casabella , Michael Joswig , Rafael Mohr
{"title":"Wronski pairs of honeycomb curves","authors":"Laura Casabella ,&nbsp;Michael Joswig ,&nbsp;Rafael Mohr","doi":"10.1016/j.jsc.2025.102528","DOIUrl":"10.1016/j.jsc.2025.102528","url":null,"abstract":"<div><div>We study certain generic systems of real polynomial equations associated with triangulations of convex polytopes and investigate their number of real solutions. Our main focus is set on pairs of plane algebraic curves which form a so-called Wronski system. The computational tasks arising in the analysis of such Wronski pairs lead us to the frontiers of current computer algebra algorithms and their implementations, both via Gröbner bases and numerical algebraic geometry.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102528"},"PeriodicalIF":1.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145580224","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
Equi-affine minimal-degree moving frames for polynomial curves 多项式曲线的等仿射最小度运动框架
IF 1.1 4区 数学
Journal of Symbolic Computation Pub Date : 2026-07-01 Epub Date: 2025-11-17 DOI: 10.1016/j.jsc.2025.102530
Hoon Hong, Irina A. Kogan
{"title":"Equi-affine minimal-degree moving frames for polynomial curves","authors":"Hoon Hong,&nbsp;Irina A. Kogan","doi":"10.1016/j.jsc.2025.102530","DOIUrl":"10.1016/j.jsc.2025.102530","url":null,"abstract":"<div><div>We develop a theory and an algorithm for constructing minimal-degree polynomial moving frames for polynomial curves in an affine space. The algorithm is equivariant under volume-preserving affine transformations of the ambient space and parameter shifts. We show that any matrix-completion algorithm can be turned into an equivariant moving frame algorithm via an equivariantization procedure that we develop. We prove that if a matrix-completion algorithm is of minimal degree, so is the resulting equivariant moving frame algorithm. We propose a novel minimal-degree matrix-completion algorithm, complementing the existing body of literature on this topic.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102530"},"PeriodicalIF":1.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145625260","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
Satisfiability modulo theories for verifying MILP certificates 验证MILP证书的可满足模理论
IF 1.1 4区 数学
Journal of Symbolic Computation Pub Date : 2026-07-01 Epub Date: 2025-12-29 DOI: 10.1016/j.jsc.2025.102543
Kenan Wood , Runtian Zhou , Haoze Wu , Hammurabi Mendes , Jonad Pulaj
{"title":"Satisfiability modulo theories for verifying MILP certificates","authors":"Kenan Wood ,&nbsp;Runtian Zhou ,&nbsp;Haoze Wu ,&nbsp;Hammurabi Mendes ,&nbsp;Jonad Pulaj","doi":"10.1016/j.jsc.2025.102543","DOIUrl":"10.1016/j.jsc.2025.102543","url":null,"abstract":"<div><div>Correctness of results from mixed-integer linear programming (MILP) solvers is critical, particularly in the context of applications such as hardware verification, compiler optimization, or machine-assisted theorem proving. To this end, VIPR 1.0 is the first recently proposed general certificate format for answers produced by MILP solvers. We design a schema to encode VIPR's inference rules as a ground formula that completely characterizes the validity of the algorithmic check, removing any ambiguities and imprecisions present in the specification. We formally verify the correctness of our schema at the logical level using <span>Why3</span>'s automated deductive logic framework. Furthermore, we implement a checker for VIPR certificates by expressing our formally verified ground formula with the Satisfiability Modulo Theory Library (SMT-LIB) and check its validity. Our approach is solver-agnostic, and we test its viability using benchmark instances found in the literature.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102543"},"PeriodicalIF":1.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145924765","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
Stratification of projection maps from toric varieties 环缘植物投影图的分层
IF 1.1 4区 数学
Journal of Symbolic Computation Pub Date : 2026-07-01 Epub Date: 2025-12-29 DOI: 10.1016/j.jsc.2025.102547
Boulos El Hilany , Martin Helmer , Elias Tsigaridas
{"title":"Stratification of projection maps from toric varieties","authors":"Boulos El Hilany ,&nbsp;Martin Helmer ,&nbsp;Elias Tsigaridas","doi":"10.1016/j.jsc.2025.102547","DOIUrl":"10.1016/j.jsc.2025.102547","url":null,"abstract":"<div><div>We define a polyhedral version of a stratification for projection maps that applies to any complex or real toric variety and show that it yields similarly desirable properties to the classical map stratification of a proper map. Our results are constructive and give rise to a method for associating the Whitney strata of the projection to the faces of the polytope of the corresponding toric variety. For all the examples we consider, our resulting algorithm outperforms known general purpose methods, e.g., Helmer and Nanda (FoCM, 2022), and Đinh and Jelonek (DCG, 2021), for computing map stratifications.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102547"},"PeriodicalIF":1.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145883344","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
Algebraic and algorithmic methods for computing polynomial loop invariants 计算多项式循环不变量的代数和算法方法
IF 1.1 4区 数学
Journal of Symbolic Computation Pub Date : 2026-07-01 Epub Date: 2025-12-31 DOI: 10.1016/j.jsc.2025.102551
Erdenebayar Bayarmagnai , Fatemeh Mohammadi , Rémi Prébet
{"title":"Algebraic and algorithmic methods for computing polynomial loop invariants","authors":"Erdenebayar Bayarmagnai ,&nbsp;Fatemeh Mohammadi ,&nbsp;Rémi Prébet","doi":"10.1016/j.jsc.2025.102551","DOIUrl":"10.1016/j.jsc.2025.102551","url":null,"abstract":"<div><div>Loop invariants are properties of a program loop that hold both before and after each iteration of the loop. They are often used to verify programs and ensure that algorithms consistently produce correct results during execution. Consequently, generating invariants becomes a crucial task for loops. We specifically focus on polynomial loops, where both the loop conditions and the assignments within the loop are expressed as polynomials. Although computing polynomial invariants for general loops is undecidable, efficient algorithms have been developed for certain classes of loops. For instance, when all assignments within a while loop involve linear polynomials, the loop becomes solvable. In this work, we study the more general case, where the polynomials can have arbitrary degrees.</div><div>Using tools from algebraic geometry, we present two algorithms designed to generate all polynomial invariants within a given vector subspace, for a branching loop with nondeterministic conditional statements. These algorithms combine linear algebraic subroutines with computations on polynomial ideals. They differ depending on whether the initial values of the loop variables are specified or treated as parameters. Additionally, we present a much more efficient algorithm for generating polynomial invariants of a specific form, applicable to all initial values. This algorithm avoids expensive ideal computations.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102551"},"PeriodicalIF":1.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145924766","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
Explicit desingularisation of Kummer surfaces in characteristic two via specialisation 特征二的Kummer曲面通过专门化的显式去具体化
IF 1.1 4区 数学
Journal of Symbolic Computation Pub Date : 2026-07-01 Epub Date: 2025-12-04 DOI: 10.1016/j.jsc.2025.102541
Alvaro Gonzalez-Hernandez
{"title":"Explicit desingularisation of Kummer surfaces in characteristic two via specialisation","authors":"Alvaro Gonzalez-Hernandez","doi":"10.1016/j.jsc.2025.102541","DOIUrl":"10.1016/j.jsc.2025.102541","url":null,"abstract":"<div><div>We study the birational geometry of the Kummer surfaces associated to the Jacobian varieties of genus two curves, with a particular focus on fields of characteristic two. In order to do so, we explicitly compute a projective embedding of the Jacobian of a general genus two curve and, from this, we construct its associated Kummer surface. This explicit construction produces a model for desingularised Kummer surfaces over any field of characteristic not two, and specialising these equations to characteristic two provides a model of a partial desingularisation. Adapting the classic description of the Picard lattice in terms of tropes, we also describe how to explicitly find completely desingularised models of Kummer surfaces whenever the <em>p</em>-rank is not zero. In the final section of this paper, we compute an example of a Kummer surface with everywhere good reduction over a quadratic number field, and draw connections between the models we computed and a criterion that determines when a Kummer surface has good reduction at two.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102541"},"PeriodicalIF":1.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145839834","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
Postulation of lines in P3 revisited 重新考虑P3中的行假设
IF 1.1 4区 数学
Journal of Symbolic Computation Pub Date : 2026-07-01 Epub Date: 2025-11-17 DOI: 10.1016/j.jsc.2025.102525
Marcin Dumnicki , Mikołaj Le Van , Grzegorz Malara , Tomasz Szemberg , Justyna Szpond , Halszka Tutaj-Gasińska
{"title":"Postulation of lines in P3 revisited","authors":"Marcin Dumnicki ,&nbsp;Mikołaj Le Van ,&nbsp;Grzegorz Malara ,&nbsp;Tomasz Szemberg ,&nbsp;Justyna Szpond ,&nbsp;Halszka Tutaj-Gasińska","doi":"10.1016/j.jsc.2025.102525","DOIUrl":"10.1016/j.jsc.2025.102525","url":null,"abstract":"<div><div>The purpose of the present note is to provide a new proof of the well-known result due to Hartshorne and Hirschowitz to the effect that general lines in projective spaces have <em>good postulation</em>. Our approach uses specialization to a hyperplane and thus opens a door to study postulation of general codimension 2 linear subspaces in projective spaces.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102525"},"PeriodicalIF":1.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145555331","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
Generalised Buchberger and Schreyer algorithms for strongly discrete coherent rings 强离散相干环的广义Buchberger和Schreyer算法
IF 1.1 4区 数学
Journal of Symbolic Computation Pub Date : 2026-07-01 Epub Date: 2025-11-14 DOI: 10.1016/j.jsc.2025.102526
Henri Lombardi , Stefan Neuwirth , Ihsen Yengui
{"title":"Generalised Buchberger and Schreyer algorithms for strongly discrete coherent rings","authors":"Henri Lombardi ,&nbsp;Stefan Neuwirth ,&nbsp;Ihsen Yengui","doi":"10.1016/j.jsc.2025.102526","DOIUrl":"10.1016/j.jsc.2025.102526","url":null,"abstract":"<div><div>Let <em>M</em> be a finitely generated submodule of a free module over a multivariate polynomial ring with coefficients in a discrete coherent ring. We prove that its module <span><math><mrow><mi>MLT</mi></mrow><mo>(</mo><mi>M</mi><mo>)</mo></math></span> of leading terms is countably generated and give an algorithm for computing explicitly a generating set. This result is also useful when <span><math><mrow><mi>MLT</mi></mrow><mo>(</mo><mi>M</mi><mo>)</mo></math></span> is not finitely generated. Suppose that the base ring is strongly discrete coherent. We provide a Buchberger-like algorithm and prove that it converges if and only if the module of leading terms is finitely generated. We also state a constructive version of Hilbert's syzygy theorem by following Schreyer's method.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"135 ","pages":"Article 102526"},"PeriodicalIF":1.1,"publicationDate":"2026-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145625259","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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