Pub Date : 2025-10-09DOI: 10.1016/j.jsc.2025.102519
Rafael Stekolshchik
<div><div>The concept of <em>special</em> and <em>extraspecial pairs</em> of roots was introduced by <span><span>Carter (1989)</span></span> to calculate structure constants of simple Lie algebras. Let <span><math><mo>{</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>}</mo></math></span> be a special pair of roots for which the structure constant <span><math><mi>N</mi><mo>(</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span> is sought, and let <span><math><mo>{</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>}</mo></math></span> be the extraspecial pair of roots corresponding to <span><math><mo>{</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>}</mo></math></span>. In this paper, we introduce the notion of a <em>quartet</em>, the ordered set <span><math><mo>{</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>}</mo></math></span> consisting of the special and corresponding extraspecial pairs. The classification of quartets makes it possible to simplify the formulas for calculations of the structure constants. Quartet <span><math><mi>q</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>}</mo></math></span> is said to be a <em>mono-quartet</em> if vectors <span><math><mi>s</mi><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><mi>r</mi><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> are not roots at the same time. Quartet <em>q</em> is said to be <em>simple</em> if it satisfies the following properties: if <span><math><mi>s</mi><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> (resp. <span><math><mi>r</mi><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>) is a root, then <span><math><mo>|</mo><mi>s</mi><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>|</mo><mo>=</mo><mo>|</mo><mi>s</mi><mo>|</mo></math></span> (resp. <span><math><mo>|</mo><mi>r</mi><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>|</mo><mo>=</mo><mo>|</mo><mi>r</mi><mo>|</mo></math></span>). It is shown that for the simple Lie algebra of type <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> (resp. type <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>) all quartets are mono- and simple (resp. simple). These facts allows to simplify the well-known recurrent formulas for calculations of structure constants derived from the fundamental Carter formula (<span><span>1989</span></span>, p. 59). As a consequence, the algorithm for calculating the structural constants can be signifi
{"title":"Extraspecial pairs in the multiply laced root systems and calculating structure constants","authors":"Rafael Stekolshchik","doi":"10.1016/j.jsc.2025.102519","DOIUrl":"10.1016/j.jsc.2025.102519","url":null,"abstract":"<div><div>The concept of <em>special</em> and <em>extraspecial pairs</em> of roots was introduced by <span><span>Carter (1989)</span></span> to calculate structure constants of simple Lie algebras. Let <span><math><mo>{</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>}</mo></math></span> be a special pair of roots for which the structure constant <span><math><mi>N</mi><mo>(</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>)</mo></math></span> is sought, and let <span><math><mo>{</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>}</mo></math></span> be the extraspecial pair of roots corresponding to <span><math><mo>{</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>}</mo></math></span>. In this paper, we introduce the notion of a <em>quartet</em>, the ordered set <span><math><mo>{</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>}</mo></math></span> consisting of the special and corresponding extraspecial pairs. The classification of quartets makes it possible to simplify the formulas for calculations of the structure constants. Quartet <span><math><mi>q</mi><mo>=</mo><mo>{</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mi>r</mi><mo>,</mo><mi>s</mi><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>}</mo></math></span> is said to be a <em>mono-quartet</em> if vectors <span><math><mi>s</mi><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><mi>r</mi><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> are not roots at the same time. Quartet <em>q</em> is said to be <em>simple</em> if it satisfies the following properties: if <span><math><mi>s</mi><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> (resp. <span><math><mi>r</mi><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>) is a root, then <span><math><mo>|</mo><mi>s</mi><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>|</mo><mo>=</mo><mo>|</mo><mi>s</mi><mo>|</mo></math></span> (resp. <span><math><mo>|</mo><mi>r</mi><mo>−</mo><msub><mrow><mi>r</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>|</mo><mo>=</mo><mo>|</mo><mi>r</mi><mo>|</mo></math></span>). It is shown that for the simple Lie algebra of type <span><math><msub><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> (resp. type <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>) all quartets are mono- and simple (resp. simple). These facts allows to simplify the well-known recurrent formulas for calculations of structure constants derived from the fundamental Carter formula (<span><span>1989</span></span>, p. 59). As a consequence, the algorithm for calculating the structural constants can be signifi","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"134 ","pages":"Article 102519"},"PeriodicalIF":1.1,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145321092","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}
Pub Date : 2025-10-08DOI: 10.1016/j.jsc.2025.102520
Hoon Hong , Jiaqi Meng , Jing Yang
Subresultants of two univariate polynomials (bi-polynomial subresultants) are one of the most classic and ubiquitous objects in computational algebra and algebraic geometry. Since their introduction, several interesting relations among them were discovered. Those relations were crucial for both structural understanding and efficient computation of bi-polynomial subresultants.
Recently, the bi-polynomial subresultants have been generalized to multi-polynomial subresultants, that is, subresultants of several polynomials. They have been already used for tackling some fundamental problems in computational algebra and algebraic geometry. Thus a natural challenge arises: Generalize, if possible, the relations among bi-polynomial subresultants to multi-polynomial subresultants.
In this paper, we tackle this challenge and provide a family of relations among multi-polynomial subresultants, which elegantly generalize some of known relations among bi-polynomial subresultants.
{"title":"Relations among multi-polynomial subresultants","authors":"Hoon Hong , Jiaqi Meng , Jing Yang","doi":"10.1016/j.jsc.2025.102520","DOIUrl":"10.1016/j.jsc.2025.102520","url":null,"abstract":"<div><div>Subresultants of two univariate polynomials (bi-polynomial subresultants) are one of the most classic and ubiquitous objects in computational algebra and algebraic geometry. Since their introduction, several interesting relations among them were discovered. Those relations were crucial for both structural understanding and efficient computation of bi-polynomial subresultants.</div><div>Recently, the bi-polynomial subresultants have been generalized to multi-polynomial subresultants, that is, subresultants of several polynomials. They have been already used for tackling some fundamental problems in computational algebra and algebraic geometry. Thus a natural challenge arises: <em>Generalize, if possible, the relations among bi-polynomial subresultants to multi-polynomial subresultants.</em></div><div>In this paper, we tackle this challenge and provide a family of relations among multi-polynomial subresultants, which elegantly generalize some of known relations among bi-polynomial subresultants.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"134 ","pages":"Article 102520"},"PeriodicalIF":1.1,"publicationDate":"2025-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145321162","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}
Pub Date : 2025-10-08DOI: 10.1016/j.jsc.2025.102518
Carlos Améndola, Kamillo Ferry
A polytrope is a tropical polyhedron that is also classically convex. We study the tropical combinatorial types of polytropes associated to weighted directed acyclic graphs (DAGs). This family of polytropes arises in algebraic statistics when describing the model class of max-linear Bayesian networks. We show how the edge weights of a network directly relate to the facet structure of the corresponding polytrope. We also give a classification of polytropes from weighted DAGs at different levels of equivalence. These results give insight on the statistical problem of identifiability for a max-linear Bayesian network.
{"title":"Tropical combinatorics of max-linear Bayesian networks","authors":"Carlos Améndola, Kamillo Ferry","doi":"10.1016/j.jsc.2025.102518","DOIUrl":"10.1016/j.jsc.2025.102518","url":null,"abstract":"<div><div>A polytrope is a tropical polyhedron that is also classically convex. We study the tropical combinatorial types of polytropes associated to weighted directed acyclic graphs (DAGs). This family of polytropes arises in algebraic statistics when describing the model class of max-linear Bayesian networks. We show how the edge weights of a network directly relate to the facet structure of the corresponding polytrope. We also give a classification of polytropes from weighted DAGs at different levels of equivalence. These results give insight on the statistical problem of identifiability for a max-linear Bayesian network.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"134 ","pages":"Article 102518"},"PeriodicalIF":1.1,"publicationDate":"2025-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145321164","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}
Pub Date : 2025-09-04DOI: 10.1016/j.jsc.2025.102507
Erich L. Kaltofen , Zhi-Hong Yang
We show that the early termination algorithm in [Kaltofen and Lee, JSC, vol. 36, nr. 3–4, 2003] for interpolating a polynomial that is a linear combination of t Chebyshev polynomials of the first kind can be modified to use randomized evaluation points; Kaltofen and Lee required randomized evaluation points. Our variants work for scalar fields of any characteristic. The number of evaluations matches that of the early termination version of the Prony sparse interpolation algorithm for the standard basis of powers of the variable [Kaltofen, Lee and Lobo, Proc. ISSAC 2000].
Our interpolation algorithm can compute the term locator polynomial in field arithmetic operations while storing intermediate field elements by Heinig's Toeplitz solver with singular sections [Heinig and Rost, “Algebraic Methods for Toeplitz-like Matrices and Operators,” Birkhäuser, 1984]. We describe a slight modification for the Levinson-Durbin-Heinig algorithm that mirrors the Berlekamp-Massey algorithm for Hankel matrices.
我们证明了[Kaltofen and Lee， JSC, vol. 36, nr. 3-4, 2003]中用于插值t个第一类Chebyshev多项式线性组合的早期终止算法可以修改为使用2t+1个随机评价点；Kaltofen和Lee需要2t+2个随机评价点。我们的变体适用于任何特征的标量场。2t+1次评估的次数与变量幂标准基的proony稀疏插值算法的早期终止版本相匹配[Kaltofen， Lee和Lobo， Proc. ISSAC 2000]。我们的插值算法可以在O（t2）个字段算术运算中计算项定位多项式，同时通过Heinig的Toeplitz求解器存储O(t)个中间字段元素，具有奇异部分[Heinig和Rost，“Toeplitz-类矩阵和算子的代数方法”Birkhäuser, 1984]。我们描述了对Levinson-Durbin-Heinig算法的轻微修改，该算法反映了Hankel矩阵的Berlekamp-Massey算法。
{"title":"Early termination for sparse interpolation of polynomials in Chebyshev bases","authors":"Erich L. Kaltofen , Zhi-Hong Yang","doi":"10.1016/j.jsc.2025.102507","DOIUrl":"10.1016/j.jsc.2025.102507","url":null,"abstract":"<div><div>We show that the early termination algorithm in [Kaltofen and Lee, JSC, vol. 36, nr. 3–4, 2003] for interpolating a polynomial that is a linear combination of <em>t</em> Chebyshev polynomials of the first kind can be modified to use <span><math><mn>2</mn><mi>t</mi><mo>+</mo><mn>1</mn></math></span> randomized evaluation points; Kaltofen and Lee required <span><math><mn>2</mn><mi>t</mi><mo>+</mo><mn>2</mn></math></span> randomized evaluation points. Our variants work for scalar fields of any characteristic. The number <span><math><mn>2</mn><mi>t</mi><mo>+</mo><mn>1</mn></math></span> of evaluations matches that of the early termination version of the Prony sparse interpolation algorithm for the standard basis of powers of the variable [Kaltofen, Lee and Lobo, Proc. ISSAC 2000].</div><div>Our interpolation algorithm can compute the term locator polynomial in <span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span> field arithmetic operations while storing <span><math><mi>O</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> intermediate field elements by Heinig's Toeplitz solver with singular sections [Heinig and Rost, “Algebraic Methods for Toeplitz-like Matrices and Operators,” Birkhäuser, 1984]. We describe a slight modification for the Levinson-Durbin-Heinig algorithm that mirrors the Berlekamp-Massey algorithm for Hankel matrices.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"134 ","pages":"Article 102507"},"PeriodicalIF":1.1,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145097421","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}
Pub Date : 2025-09-04DOI: 10.1016/j.jsc.2025.102506
Thierry Combot
We consider the problem of symbolic integration of where G is rational and is a non algebraic solution of a differential equation with F rational. Substituting in the integral by , the general solution of , we have a parametrized integral . We prove that is, as a two variable function in , either differentially transcendental, or, with a good parametrization in h, there exists a linear differential operator L in h with constant coefficients, called a telescoper, such that is rational in and the h derivatives of y. This notion generalizes elementary integration. We present an algorithm to compute such telescoper given a priori bound on the order ord of L and degree N of , with complexity . For the specific foliation , a more complete algorithm without a priori bound is presented. Oppositely, non existence of telescoper is proven for a classical planar Hamiltonian system. As an application, we present an algorithm which always finds, if they exist, the Liouvillian solutions of a planar rational vector field, given a bound large enough for some notion of complexity height.
{"title":"Symbolic integration on planar differential foliations","authors":"Thierry Combot","doi":"10.1016/j.jsc.2025.102506","DOIUrl":"10.1016/j.jsc.2025.102506","url":null,"abstract":"<div><div>We consider the problem of symbolic integration of <span><math><mo>∫</mo><mi>G</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mi>d</mi><mi>x</mi></math></span> where <em>G</em> is rational and <span><math><mi>y</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is a non algebraic solution of a differential equation <span><math><msup><mrow><mi>y</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>F</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span> with <em>F</em> rational. Substituting in the integral <span><math><mi>y</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> by <span><math><mi>y</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>h</mi><mo>)</mo></math></span>, the general solution of <span><math><msup><mrow><mi>y</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>F</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span>, we have a parametrized integral <span><math><mi>I</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>h</mi><mo>)</mo></math></span>. We prove that <span><math><mi>I</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>h</mi><mo>)</mo></math></span> is, as a two variable function in <span><math><mi>x</mi><mo>,</mo><mi>h</mi></math></span>, either differentially transcendental, or, with a good parametrization in <em>h</em>, there exists a linear differential operator <em>L</em> in <em>h</em> with constant coefficients, called a telescoper, such that <span><math><mi>L</mi><mi>I</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>h</mi><mo>)</mo></math></span> is rational in <span><math><mi>x</mi><mo>,</mo><mi>y</mi></math></span> and the <em>h</em> derivatives of <em>y</em>. This notion generalizes elementary integration. We present an algorithm to compute such telescoper given a priori bound on the order ord of <em>L</em> and degree <em>N</em> of <span><math><mi>L</mi><mi>I</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>h</mi><mo>)</mo></math></span>, with complexity <span><math><mover><mrow><mi>O</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><msup><mrow><mi>N</mi></mrow><mrow><mi>ω</mi><mo>+</mo><mn>1</mn></mrow></msup><msup><mrow><mtext>ord</mtext></mrow><mrow><mi>ω</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>+</mo><mi>N</mi><msup><mrow><mtext>ord</mtext></mrow><mrow><mi>ω</mi><mo>+</mo><mn>3</mn></mrow></msup><mo>)</mo></math></span>. For the specific foliation <span><math><mi>y</mi><mo>=</mo><mi>ln</mi><mo></mo><mi>x</mi></math></span>, a more complete algorithm without a priori bound is presented. Oppositely, non existence of telescoper is proven for a classical planar Hamiltonian system. As an application, we present an algorithm which always finds, if they exist, the Liouvillian solutions of a planar rational vector field, given a bound large enough for some notion of complexity height.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"134 ","pages":"Article 102506"},"PeriodicalIF":1.1,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145049629","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}
Pub Date : 2025-09-04DOI: 10.1016/j.jsc.2025.102508
Xin Chen, Weiping Wang
In this paper, by applying the general Fourier-Legendre series expansion, we establish four general series transformations, and obtain a range of relations between the parametric Apéry-type series and the double sums of products of multiple harmonic sums (MHSs) or multiple t-harmonic sums (MtSs) from the Fourier-Legendre series expansions of the complete elliptic integrals of the first and second kind as well as two special expansions provided recently in the literature. By establishing the linearization theorem for the double sums of products above, and using the methods of partial fraction decomposition and transformation of summations, we show that these parametric Apéry-type series are expressible in terms of some elementary series involving MHSs and MtSs, and finally reducible, with an extra factor , to linear combinations of alternating multiple zeta values and colored multiple zeta values of level four. By specifying the parameters, we determine the evaluations of many special Apéry-type series.
{"title":"Apéry-type series via colored multiple zeta values and Fourier-Legendre series expansions","authors":"Xin Chen, Weiping Wang","doi":"10.1016/j.jsc.2025.102508","DOIUrl":"10.1016/j.jsc.2025.102508","url":null,"abstract":"<div><div>In this paper, by applying the general Fourier-Legendre series expansion, we establish four general series transformations, and obtain a range of relations between the parametric Apéry-type series and the double sums of products of multiple harmonic sums (MHSs) or multiple <em>t</em>-harmonic sums (MtSs) from the Fourier-Legendre series expansions of the complete elliptic integrals of the first and second kind as well as two special expansions provided recently in the literature. By establishing the linearization theorem for the double sums of products above, and using the methods of partial fraction decomposition and transformation of summations, we show that these parametric Apéry-type series are expressible in terms of some elementary series involving MHSs and MtSs, and finally reducible, with an extra factor <span><math><mn>1</mn><mo>/</mo><mi>π</mi></math></span>, to linear combinations of alternating multiple zeta values and colored multiple zeta values of level four. By specifying the parameters, we determine the evaluations of many special Apéry-type series.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"134 ","pages":"Article 102508"},"PeriodicalIF":1.1,"publicationDate":"2025-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145050267","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}
Pub Date : 2025-08-29DOI: 10.1016/j.jsc.2025.102505
Teresa Krick , Agnes Szanto
In this note we explicit the notion of Hermite interpolant of a multivariate symmetric polynomial, generalizing the notion of Lagrange interpolant to the case when there are roots coalescence, an extension of the results on the symmetric Hermite interpolation basis by M.-F. Roy and A. Szpirglas included in (Roy and Szpirglas, 2020).
本文给出了多元对称多项式的Hermite插值的概念，将Lagrange插值的概念推广到有根合并的情况，并在对称Hermite插值的基础上推广了m - f。（Roy and Szpirglas, 2020）。
{"title":"A note on the multivariate symmetric Hermite interpolant","authors":"Teresa Krick , Agnes Szanto","doi":"10.1016/j.jsc.2025.102505","DOIUrl":"10.1016/j.jsc.2025.102505","url":null,"abstract":"<div><div>In this note we explicit the notion of Hermite interpolant of a multivariate symmetric polynomial, generalizing the notion of Lagrange interpolant to the case when there are roots coalescence, an extension of the results on the symmetric Hermite interpolation basis by M.-F. Roy and A. Szpirglas included in (<span><span>Roy and Szpirglas, 2020</span></span>).</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"134 ","pages":"Article 102505"},"PeriodicalIF":1.1,"publicationDate":"2025-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145011140","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}
Pub Date : 2025-08-21DOI: 10.1016/j.jsc.2025.102489
Zuan Liu , Zihao Qi , Yufei Qin , Guodong Zhou
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 B can be generated on an algebra A similar to a free algebra generated on a set. If A has a Gröbner-Shirshov basis G and if the linear operators Ω satisfy a set Φ of operator identities, it is natural to ask when the union is a Gröbner-Shirshov basis of B. A previous paper answers this question affirmatively under a mild condition, and thereby obtains a canonical linear basis of B.
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.
{"title":"Gröbner-Shirshov bases for free multi-operated algebras over algebras","authors":"Zuan Liu , Zihao Qi , Yufei Qin , 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":"2025-08-21","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}
We study the relation between the bigraded Castelnuovo-Mumford regularity of a bihomogeneous ideal I in the coordinate ring of the product of two projective spaces and the bidegrees of a Gröbner basis of I 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 I. 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 I.
{"title":"Bigraded Castelnuovo-Mumford regularity and Gröbner bases","authors":"Matías Bender , Laurent Busé , Carles Checa , 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":"2025-08-19","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}
Pub Date : 2025-08-14DOI: 10.1016/j.jsc.2025.102486
Devlin Mallory , Mahrud Sayrafi
We describe and prove correctness of two practical algorithms for finding indecomposable summands of finitely generated modules over a finitely generated k-algebra R. 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 R is local and k 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.
{"title":"Computing direct sum decompositions","authors":"Devlin Mallory , 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":"2025-08-14","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}
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