Journal of Symbolic Computation最新文献

{"title":"Early termination for sparse interpolation of polynomials in Chebyshev bases","authors":"Erich L. Kaltofen ,&nbsp;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}

{"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}

{"title":"Apéry-type series via colored multiple zeta values and Fourier-Legendre series expansions","authors":"Xin Chen,&nbsp;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}

{"title":"A note on the multivariate symmetric Hermite interpolant","authors":"Teresa Krick ,&nbsp;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}

{"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":"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}

{"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":"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}

{"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":"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}

{"title":"Computations of algebraic modular forms associated with the definite quaternion algebra of discriminant 2","authors":"Hiroyuki Ochiai ,&nbsp;Satoshi Wakatsuki ,&nbsp;Shun'ichi Yokoyama","doi":"10.1016/j.jsc.2025.102485","DOIUrl":"10.1016/j.jsc.2025.102485","url":null,"abstract":"<div><div>In this paper, we present an algorithm to compute a basis of the space of algebraic modular forms on the maximal order of the definite quaternion algebra of discriminant 2, and provide a database of such bases. A main application of our database is to obtain congruence relations of algebraic modular forms, which lead non-vanishing theorems for prime twists of modular <em>L</em>-functions.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"133 ","pages":"Article 102485"},"PeriodicalIF":1.1,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144886193","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Solvable and nilpotent matroids: Realizability and irreducible decomposition of their associated varieties","authors":"Emiliano Liwski,&nbsp;Fatemeh Mohammadi","doi":"10.1016/j.jsc.2025.102484","DOIUrl":"10.1016/j.jsc.2025.102484","url":null,"abstract":"<div><div>We introduce the families of solvable and nilpotent matroids, examining their realization spaces, closures, and associated matroid and circuit varieties. We study their realizability, as well as the irreducible decomposition of their associated matroid and circuit varieties. Additionally, we describe a finite generating set for the corresponding ideals, considered up to radical. We establish sufficient conditions for both the realizability of these matroids and the irreducibility of their associated varieties. Specifically, we establish the realizability and irreducibility of matroid varieties associated with nilpotent matroids and prove the irreducibility of matroid varieties arising from certain classes of solvable paving matroids. Additionally, we analyze the defining polynomial equations of these varieties using Grassmann-Cayley algebra and geometric liftability techniques. Furthermore, we provide a complete generating set for the matroid ideals associated with forest configurations.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"133 ","pages":"Article 102484"},"PeriodicalIF":1.1,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144886195","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On semi-local decomposition","authors":"Ming-Deh A. Huang","doi":"10.1016/j.jsc.2025.102483","DOIUrl":"10.1016/j.jsc.2025.102483","url":null,"abstract":"<div><div>We consider semi-local polynomial systems and their decomposition. A semi-local polynomial system defines a global polynomial map that is the product of local polynomial maps disguised by global linear isomorphisms. We characterize a subclass of semi-local polynomial systems which can be efficiently decomposed into local systems.</div></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"133 ","pages":"Article 102483"},"PeriodicalIF":1.1,"publicationDate":"2025-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144886196","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}