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Computing direct sum decompositions 计算直接和分解
IF 1.1 4区 数学
Journal of Symbolic Computation Pub Date : 2026-03-01 Epub Date: 2025-08-14 DOI: 10.1016/j.jsc.2025.102486
Devlin Mallory , Mahrud Sayrafi
{"title":"Computing direct sum decompositions","authors":"Devlin Mallory ,&nbsp;Mahrud Sayrafi","doi":"10.1016/j.jsc.2025.102486","DOIUrl":"10.1016/j.jsc.2025.102486","url":null,"abstract":"<div><div>We describe and prove correctness of two practical algorithms for finding indecomposable summands of finitely generated modules over a finitely generated <em>k</em>-algebra <em>R</em>. The first algorithm applies in the (multi)graded case, which enables the computation of indecomposable summands of coherent sheaves on subvarieties of toric varieties (in particular, for varieties embedded in projective space); the second algorithm applies when <em>R</em> is local and <em>k</em> is a finite field, opening the door to computing decompositions in singularity theory. We also present multiple examples, including some which present previously unknown phenomena regarding the behavior of summands of Frobenius pushforwards (including in the non-graded case) and syzygies over Artinian rings.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"133 ","pages":"Article 102486"},"PeriodicalIF":1.1,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144903701","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
Subalgebra and Khovanskii bases equivalence 子代数与Khovanskii基等价
IF 1.1 4区 数学
Journal of Symbolic Computation Pub Date : 2026-03-01 Epub Date: 2025-08-06 DOI: 10.1016/j.jsc.2025.102480
Colin Alstad , Michael Burr , Oliver Clarke , Timothy Duff
{"title":"Subalgebra and Khovanskii bases equivalence","authors":"Colin Alstad ,&nbsp;Michael Burr ,&nbsp;Oliver Clarke ,&nbsp;Timothy Duff","doi":"10.1016/j.jsc.2025.102480","DOIUrl":"10.1016/j.jsc.2025.102480","url":null,"abstract":"<div><div>We study a partial correspondence between two previously-studied analogues of Gröbner bases in the setting of algebras: namely subalgebra bases for quotients of polynomial rings and Khovanskii bases for valued algebras and domains. Our main motivation is to apply the concrete and computational aspects of subalgebra bases for quotient rings to the abstract theory of Khovanskii bases. Our perspective is that most interesting examples of Khovanskii bases can also be realized as subalgebra bases and vice-versa. As part of this correspondence, we extend the theory of subalgebra bases for quotients of polynomial rings to infinitely generated polynomial algebras and study conditions which make this theory effective. We also provide a computation of Newton-Okounkov bodies from the data of subalgebra bases for quotient rings, which illustrates how interpreting Khovanskii bases as subalgebra bases makes them amenable to existing computer algebra tools.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"133 ","pages":"Article 102480"},"PeriodicalIF":1.1,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144886194","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
Bigraded Castelnuovo-Mumford regularity and Gröbner bases 升级Castelnuovo-Mumford规则和Gröbner基地
IF 1.1 4区 数学
Journal of Symbolic Computation Pub Date : 2026-03-01 Epub Date: 2025-08-19 DOI: 10.1016/j.jsc.2025.102487
Matías Bender , Laurent Busé , Carles Checa , Elias Tsigaridas
{"title":"Bigraded Castelnuovo-Mumford regularity and Gröbner bases","authors":"Matías Bender ,&nbsp;Laurent Busé ,&nbsp;Carles Checa ,&nbsp;Elias Tsigaridas","doi":"10.1016/j.jsc.2025.102487","DOIUrl":"10.1016/j.jsc.2025.102487","url":null,"abstract":"<div><div>We study the relation between the bigraded Castelnuovo-Mumford regularity of a bihomogeneous ideal <em>I</em> in the coordinate ring of the product of two projective spaces and the bidegrees of a Gröbner basis of <em>I</em> with respect to the degree reverse lexicographical monomial order in generic coordinates. For the single-graded case, Bayer and Stillman unraveled all aspects of this relationship forty years ago and these results led to complexity estimates for computations with Gröbner bases. We build on this work to introduce a bounding region of the bidegrees of minimal generators of bihomogeneous Gröbner bases for <em>I</em>. We also use this region to certify the presence of some minimal generators close to its boundary. Finally, we show that, up to a certain shift, this region is related to the bigraded Castelnuovo-Mumford regularity of <em>I</em>.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"133 ","pages":"Article 102487"},"PeriodicalIF":1.1,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144880233","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
Gröbner-Shirshov bases for free multi-operated algebras over algebras Gröbner-Shirshov代数上自由多操作代数的基
IF 1.1 4区 数学
Journal of Symbolic Computation Pub Date : 2026-03-01 Epub Date: 2025-08-21 DOI: 10.1016/j.jsc.2025.102489
Zuan Liu , Zihao Qi , Yufei Qin , Guodong Zhou
{"title":"Gröbner-Shirshov bases for free multi-operated algebras over algebras","authors":"Zuan Liu ,&nbsp;Zihao Qi ,&nbsp;Yufei Qin ,&nbsp;Guodong Zhou","doi":"10.1016/j.jsc.2025.102489","DOIUrl":"10.1016/j.jsc.2025.102489","url":null,"abstract":"<div><div>Operated algebras have recently attracted considerable attention, as they unify various structures such as differential algebras and Rota-Baxter algebras. An Ω-operated algebra is an associative algebra equipped with a set Ω of linear operators which might satisfy certain operator identities such as the Leibniz rule. A free Ω-operated algebra <em>B</em> can be generated on an algebra <em>A</em> similar to a free algebra generated on a set. If <em>A</em> has a Gröbner-Shirshov basis <em>G</em> and if the linear operators Ω satisfy a set Φ of operator identities, it is natural to ask when the union <span><math><mi>G</mi><mo>∪</mo><mi>Φ</mi></math></span> is a Gröbner-Shirshov basis of <em>B</em>. A previous paper answers this question affirmatively under a mild condition, and thereby obtains a canonical linear basis of <em>B</em>.</div><div>In this paper, we answer this question in the general case of multiple linear operators. As applications we get operated Gröbner-Shirshov bases for free differential Rota-Baxter algebras and free integro-differential algebras over algebras as well as their linear bases. One of the key technical difficulties is to introduce new monomial orders for the case of two operators, which might be of independent interest.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"133 ","pages":"Article 102489"},"PeriodicalIF":1.1,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144908024","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
Slices of stable polynomials and connections to the Grace-Walsh-Szegő theorem 稳定多项式的切片和格雷斯-沃尔什-塞格尔定理的联系
IF 1.1 4区 数学
Journal of Symbolic Computation Pub Date : 2026-03-01 Epub Date: 2025-08-14 DOI: 10.1016/j.jsc.2025.102488
Sebastian Debus , Cordian Riener , Robin Schabert
{"title":"Slices of stable polynomials and connections to the Grace-Walsh-Szegő theorem","authors":"Sebastian Debus ,&nbsp;Cordian Riener ,&nbsp;Robin Schabert","doi":"10.1016/j.jsc.2025.102488","DOIUrl":"10.1016/j.jsc.2025.102488","url":null,"abstract":"<div><div>Univariate polynomials are called stable with respect to a domain <em>D</em> if all of their roots lie in <em>D</em>. We study linear slices of the space of stable univariate polynomials with respect to a half-plane. We show that a linear slice always contains a stable polynomial with only a few distinct roots. Subsequently, we apply these results to symmetric polynomials and varieties. We show that for varieties defined by few multiaffine symmetric polynomials, the existence of a point in <span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with few distinct coordinates is necessary and sufficient for the intersection with <span><math><msup><mrow><mi>D</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> to be non-empty. This is at the same time a generalization of the so-called degree principle to stable polynomials and a result similar to Grace-Walsh-Szegő's coincidence theorem.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"133 ","pages":"Article 102488"},"PeriodicalIF":1.1,"publicationDate":"2026-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144903702","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 duality of SONC: Advances in circuit-based certificates SONC的双重性：基于电路的证书的进展
IF 1.1 4区 数学
Journal of Symbolic Computation Pub Date : 2026-01-01 Epub Date: 2025-08-05 DOI: 10.1016/j.jsc.2025.102479
Janin Heuer, Timo de Wolff
{"title":"The duality of SONC: Advances in circuit-based certificates","authors":"Janin Heuer,&nbsp;Timo de Wolff","doi":"10.1016/j.jsc.2025.102479","DOIUrl":"10.1016/j.jsc.2025.102479","url":null,"abstract":"<div><div>The cone of sums of nonnegative circuits (SONCs) is a subset of the cone of nonnegative polynomials / exponential sums, which has been studied extensively in recent years. In this article, we construct a subset of the SONC cone which we call the DSONC cone. The DSONC cone is naturally derived from the dual SONC cone; membership can be tested via linear programming. We show that the DSONC cone is a proper, full-dimensional cone, we provide a description of its extreme rays, and collect several properties that parallel those of the SONC cone. Moreover, we show that functions in the DSONC cone cannot have real zeros, which yields that DSONC cone does not intersect the boundary of the SONC cone. Furthermore, we discuss the intersection of the DSONC cone with the SOS and SDSOS cones. Finally, we show that circuit functions in the boundary of the DSONC cone are determined by points of equilibria, which hence are the analogues to singular points in the primal SONC cone, and relate the DSONC cone to tropical geometry.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102479"},"PeriodicalIF":1.1,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144826603","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
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 : 2026-01-01 Epub 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":"2026-01-01","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
Resultants of skew polynomials over division rings 除法环上偏多项式的结果
IF 1.1 4区 数学
Journal of Symbolic Computation Pub Date : 2026-01-01 Epub Date: 2025-07-23 DOI: 10.1016/j.jsc.2025.102476
Alexis Eduardo Almendras Valdebenito , Jonathan Armando Briones Donoso , Andrea Luigi Tironi
{"title":"Resultants of skew polynomials over division rings","authors":"Alexis Eduardo Almendras Valdebenito ,&nbsp;Jonathan Armando Briones Donoso ,&nbsp;Andrea Luigi Tironi","doi":"10.1016/j.jsc.2025.102476","DOIUrl":"10.1016/j.jsc.2025.102476","url":null,"abstract":"<div><div>Let <span><math><mi>F</mi></math></span> be a division ring. We generalize some of the main well-known results about the resultant of two univariate polynomials to the more general context of an Ore extension <span><math><mi>F</mi><mo>[</mo><mi>x</mi><mo>;</mo><mi>σ</mi><mo>,</mo><mi>δ</mi><mo>]</mo></math></span>. Moreover, some algorithms and Magma programs are given as a numerical application of the main theoretical results of this paper.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102476"},"PeriodicalIF":1.1,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144724144","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
Simultaneous rational number codes: Decoding beyond half the minimum distance with multiplicities and bad primes 同时的有理数码：译码超过一半的最小距离与多重和坏素数
IF 1.1 4区 数学
Journal of Symbolic Computation Pub Date : 2026-01-01 Epub Date: 2025-08-06 DOI: 10.1016/j.jsc.2025.102481
Matteo Abbondati, Eleonora Guerrini, Romain Lebreton
{"title":"Simultaneous rational number codes: Decoding beyond half the minimum distance with multiplicities and bad primes","authors":"Matteo Abbondati,&nbsp;Eleonora Guerrini,&nbsp;Romain Lebreton","doi":"10.1016/j.jsc.2025.102481","DOIUrl":"10.1016/j.jsc.2025.102481","url":null,"abstract":"<div><div>In the previous work of <span><span>Abbondati et al. (2024)</span></span>, we extended the decoding analysis of interleaved Chinese remainder codes to simultaneous rational number codes. In this work, we build on <span><span>Abbondati et al. (2024)</span></span> by addressing two important scenarios: multiplicities and the presence of bad primes (divisors of denominators). First, we generalize previous results to multiplicity rational codes by considering modular reductions with respect to prime power moduli. Then, using hybrid analysis techniques, we extend our approach to vectors of fractions that may present bad primes.</div><div>Our contributions include: a decoding algorithm for simultaneous rational number reconstruction with multiplicities, a rigorous analysis of the algorithm's failure probability that generalizes several previous results, an extension to a hybrid model handling situations where not all errors can be assumed random, and a unified approach to handle bad primes within multiplicities. The theoretical results provide a comprehensive probabilistic analysis of reconstruction failure in these more complex scenarios, advancing the state of the art in error correction for rational number codes.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102481"},"PeriodicalIF":1.1,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144828646","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
Jacobi stability analysis for systems of ODEs with symbolic computation 二阶微分方程系统的雅可比稳定性分析
IF 1.1 4区 数学
Journal of Symbolic Computation Pub Date : 2026-01-01 Epub Date: 2025-07-25 DOI: 10.1016/j.jsc.2025.102478
Bo Huang , Dongming Wang , Xinyu Wang , Jing Yang
{"title":"Jacobi stability analysis for systems of ODEs with symbolic computation","authors":"Bo Huang ,&nbsp;Dongming Wang ,&nbsp;Xinyu Wang ,&nbsp;Jing Yang","doi":"10.1016/j.jsc.2025.102478","DOIUrl":"10.1016/j.jsc.2025.102478","url":null,"abstract":"<div><div>The classical theory of Kosambi–Cartan–Chern (KCC) developed in differential geometry provides a powerful method for analyzing the behaviors of dynamical systems. In the KCC theory, the properties of a dynamical system are described in terms of five geometrical invariants, of which the second corresponds to the so-called Jacobi stability of the system. Different from that of the Lyapunov stability that has been studied extensively in the literature, the analysis of the Jacobi stability has been investigated more recently using geometrical concepts and tools. It turns out that the existing work on the Jacobi stability analysis remains theoretical and the problem of algorithmic and symbolic treatment of Jacobi stability analysis has yet to be addressed. In this paper, we initiate our study on the problem for a class of ODE systems of arbitrary dimension and propose two algorithmic schemes using symbolic computation to check whether a nonlinear dynamical system may exhibit Jacobi stability. The first scheme, based on the construction of the complex root structure of a characteristic polynomial and on the method of quantifier elimination, is capable of detecting the existence of the Jacobi stability of the given dynamical system. The second algorithmic scheme exploits the method of semi-algebraic system solving and allows one to determine conditions on the parameters for a given dynamical system to have a prescribed number of Jacobi stable fixed points. Several examples are presented to demonstrate the effectiveness of the proposed algorithmic schemes. The computational results on Jacobi stability of these examples are further verified by numerical simulations.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"132 ","pages":"Article 102478"},"PeriodicalIF":1.1,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144766755","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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