Journal of Symbolic Computation最新文献

{"title":"Solving second order homogeneous differential equations in terms of Heun's general function","authors":"Shayea Aldossari","doi":"10.1016/j.jsc.2024.102347","DOIUrl":"10.1016/j.jsc.2024.102347","url":null,"abstract":"<div><p>In this paper, we present an algorithm that checks if a second-order differential operator <span><math><mi>L</mi><mo>∈</mo><mi>C</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>[</mo><mo>∂</mo><mo>]</mo></math></span> can be reduced to the general Heun's differential operator. The algorithm detects the parameters of the transformations in <span><math><mi>C</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>[</mo><mo>∂</mo><mo>]</mo></math></span> that transfer the general Heun's differential operator to the operator <em>L</em> whose solutions are of the form<span><span><span>(1)</span><span><math><mrow><mi>exp</mi></mrow><mo>(</mo><mo>∫</mo><mi>r</mi><mspace></mspace><mi>d</mi><mi>x</mi><mo>)</mo><mo>⋅</mo><mrow><mi>HeunG</mi></mrow><mo>(</mo><mi>a</mi><mo>,</mo><mspace></mspace><mi>q</mi><mo>;</mo><mi>α</mi><mo>,</mo><mi>β</mi><mo>,</mo><mi>γ</mi><mo>,</mo><mi>δ</mi><mo>;</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo><mo>,</mo></math></span></span></span> where <span><math><mo>{</mo><mi>α</mi><mo>,</mo><mspace></mspace><mi>β</mi><mo>,</mo><mspace></mspace><mi>δ</mi><mo>,</mo><mspace></mspace><mi>γ</mi><mo>}</mo><mo>∈</mo><mi>Q</mi><mo>∖</mo><mi>Z</mi></math></span>, the functions <span><math><mi>r</mi><mo>,</mo><mspace></mspace><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><mi>C</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>, and <span><math><mi>C</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></math></span> is a subfield of <span><math><mi>C</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> of index 2 or 3 or <span><math><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi>a</mi><mspace></mspace><msup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><mi>b</mi></mrow><mrow><mi>c</mi><mspace></mspace><msup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><mi>d</mi></mrow></mfrac></math></span> for some <em>n</em> in <span><math><mi>N</mi><mo>∖</mo><mo>{</mo><mn>1</mn><mo>}</mo></math></span>.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"126 ","pages":"Article 102347"},"PeriodicalIF":0.6,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141401722","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":"The deviation on cranks of partitions","authors":"Julia Q.D. Du","doi":"10.1016/j.jsc.2024.102346","DOIUrl":"10.1016/j.jsc.2024.102346","url":null,"abstract":"<div><p>In this paper, we present an algorithm to compute the deviation of the cranks from the average by using the theory of modular forms and Jacobi forms. Then applying the Ramanujan-type algorithm developed by Chen, Du and Zhao to each term in the expression of the deviation, we can derive the corresponding dissection formulas. As applications, we revisit the deviation of the cranks modulo 5 and 7, which were given by Garvan, and Mortenson, and also obtain the deviation of the cranks modulo 9 and 14.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"126 ","pages":"Article 102346"},"PeriodicalIF":0.6,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141395493","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 post-quantum key exchange protocol from the intersection of conics","authors":"Alberto Alzati ,&nbsp;Daniele Di Tullio,&nbsp;Manoj Gyawali,&nbsp;Alfonso Tortora","doi":"10.1016/j.jsc.2024.102343","DOIUrl":"https://doi.org/10.1016/j.jsc.2024.102343","url":null,"abstract":"<div><p>In this paper we present a key exchange protocol in which Alice and Bob have secret keys given by two conics embedded in a large ambient space by means of the Veronese embedding and public keys given by hyperplanes containing the embedded curves. Both of them construct some common invariants given by the intersection of two conics.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"126 ","pages":"Article 102343"},"PeriodicalIF":0.6,"publicationDate":"2024-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0747717124000476/pdfft?md5=c1a759b0f1c037e879260d12c7a34bcb&pid=1-s2.0-S0747717124000476-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141483104","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Submodule approach to creative telescoping","authors":"Mark van Hoeij","doi":"10.1016/j.jsc.2024.102342","DOIUrl":"https://doi.org/10.1016/j.jsc.2024.102342","url":null,"abstract":"<div><p>This paper proposes ideas to speed up the process of creative telescoping, particularly when the telescoper is reducible. One can interpret telescoping as computing an annihilator <span><math><mi>L</mi><mo>∈</mo><mi>D</mi></math></span> for an element <em>m</em> in a <em>D</em>-module <em>M</em>. The main idea in this paper is to look for submodules of <em>M</em>. If <em>N</em> is a non-trivial submodule of <em>M</em>, constructing the minimal annihilator <em>R</em> of the image of <em>m</em> in <span><math><mi>M</mi><mo>/</mo><mi>N</mi></math></span> gives a right-factor of <em>L</em> in <em>D</em>. Then <span><math><mi>L</mi><mo>=</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>′</mo></mrow></msup><mi>R</mi></math></span> where the left-factor <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> is the telescoper of <span><math><mi>R</mi><mo>(</mo><mi>m</mi><mo>)</mo><mo>∈</mo><mi>N</mi></math></span>. To expedite computing <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>, compute the action of <em>D</em> on a natural basis of <em>N</em>, then obtain <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> with a cyclic vector computation.</p><p>The next main idea is to construct submodules from automorphisms, if we can find some. An automorphism with distinct eigenvalues can be used to decompose <em>N</em> as a direct sum <span><math><msub><mrow><mi>N</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⊕</mo><mo>⋯</mo><mo>⊕</mo><msub><mrow><mi>N</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>. Then <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> is the LCLM (Least Common Left Multiple) of <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> where <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is the telescoper of the projection of <span><math><mi>R</mi><mo>(</mo><mi>m</mi><mo>)</mo></math></span> on <span><math><msub><mrow><mi>N</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span>. An LCLM can greatly increase the degrees of coefficients, so <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> and <em>L</em> can be much larger expressions than the factors <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> and <em>R</em>. Examples show that computing each factor <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> and <em>R</em> separately can save a lot of CPU time compared to computing <em>L</em> in expanded form with standard creative telescoping.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"126 ","pages":"Article 102342"},"PeriodicalIF":0.6,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141483105","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":"Reduction-based creative telescoping for P-recursive sequences via integral bases","authors":"Shaoshi Chen ,&nbsp;Lixin Du ,&nbsp;Manuel Kauers ,&nbsp;Rong-Hua Wang","doi":"10.1016/j.jsc.2024.102341","DOIUrl":"https://doi.org/10.1016/j.jsc.2024.102341","url":null,"abstract":"<div><p>We propose a way to split a given bivariate P-recursive sequence into a summable part and a non-summable part in such a way that the non-summable part is minimal in some sense. This decomposition gives rise to a new reduction-based creative telescoping algorithm based on the concept of integral bases.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"126 ","pages":"Article 102341"},"PeriodicalIF":0.7,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0747717124000452/pdfft?md5=77af70a9370b6d4cc1266d4be05c2fe3&pid=1-s2.0-S0747717124000452-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141325194","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Solving equations using Khovanskii bases","authors":"Barbara Betti ,&nbsp;Marta Panizzut ,&nbsp;Simon Telen","doi":"10.1016/j.jsc.2024.102340","DOIUrl":"10.1016/j.jsc.2024.102340","url":null,"abstract":"<div><p>We develop a new eigenvalue method for solving structured polynomial equations over any field. The equations are defined on a projective algebraic variety which admits a rational parameterization by a Khovanskii basis, e.g., a Grassmannian in its Plücker embedding. This generalizes established algorithms for toric varieties, and introduces the effective use of Khovanskii bases in computer algebra. We investigate regularity questions and discuss several applications.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"126 ","pages":"Article 102340"},"PeriodicalIF":0.7,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0747717124000440/pdfft?md5=9e6933be1fe9c296b695fd040a1b4944&pid=1-s2.0-S0747717124000440-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141195382","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Reduction-based creative telescoping for definite summation of D-finite functions","authors":"Hadrien Brochet,&nbsp;Bruno Salvy","doi":"10.1016/j.jsc.2024.102329","DOIUrl":"10.1016/j.jsc.2024.102329","url":null,"abstract":"<div><p>Creative telescoping is an algorithmic method initiated by Zeilberger to compute definite sums by synthesizing summands that telescope, called certificates. We describe a creative telescoping algorithm that computes telescopers for definite sums of D-finite functions as well as the associated certificates in a compact form. The algorithm relies on a discrete analogue of the generalized Hermite reduction, or equivalently, a generalization of the Abramov-Petkovšek reduction. We provide a Maple implementation with good timings on a variety of examples.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"125 ","pages":"Article 102329"},"PeriodicalIF":0.7,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140835914","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":"Hypergeometric-type sequences","authors":"Bertrand Teguia Tabuguia","doi":"10.1016/j.jsc.2024.102328","DOIUrl":"https://doi.org/10.1016/j.jsc.2024.102328","url":null,"abstract":"<div><p>We introduce hypergeometric-type sequences. They are linear combinations of interlaced hypergeometric sequences (of arbitrary interlacements). We prove that they form a subring of the ring of holonomic sequences. An interesting family of sequences in this class are those defined by trigonometric functions with linear arguments in the index and <em>π</em>, such as Chebyshev polynomials, <span><math><msub><mrow><mo>(</mo><msup><mrow><mi>sin</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>⁡</mo><mrow><mo>(</mo><mi>n</mi><mspace></mspace><mi>π</mi><mo>/</mo><mn>4</mn><mo>)</mo></mrow><mo>⋅</mo><mi>cos</mi><mo>⁡</mo><mrow><mo>(</mo><mi>n</mi><mspace></mspace><mi>π</mi><mo>/</mo><mn>6</mn><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msub></math></span>, and compositions like <span><math><msub><mrow><mo>(</mo><mi>sin</mi><mo>⁡</mo><mrow><mo>(</mo><mi>cos</mi><mo>⁡</mo><mo>(</mo><mi>n</mi><mi>π</mi><mo>/</mo><mn>3</mn><mo>)</mo><mi>π</mi><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msub></math></span>.</p><p>We describe an algorithm that computes a hypergeometric-type normal form of a given holonomic <em>n</em>th term whenever it exists. Our implementation enables us to generate several identities for terms defined via trigonometric functions.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"125 ","pages":"Article 102328"},"PeriodicalIF":0.7,"publicationDate":"2024-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0747717124000324/pdfft?md5=51632bbf215cfdc91e40412d4a4946e1&pid=1-s2.0-S0747717124000324-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140823031","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Computations of Gromov–Witten invariants of toric varieties","authors":"Giosuè Muratore","doi":"10.1016/j.jsc.2024.102330","DOIUrl":"https://doi.org/10.1016/j.jsc.2024.102330","url":null,"abstract":"<div><p>We present the Julia package <span>ToricAtiyahBott.jl</span>, providing an easy way to perform the Atiyah–Bott formula on the moduli space of genus 0 stable maps <span><math><msub><mrow><mover><mrow><mi>M</mi></mrow><mo>‾</mo></mover></mrow><mrow><mn>0</mn><mo>,</mo><mi>m</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>,</mo><mi>β</mi><mo>)</mo></math></span> where <em>X</em> is any smooth projective toric variety, and <em>β</em> is any effective 1-cycle. The list of the supported cohomological cycles contains the most common ones, and it is extensible. We provide a detailed explanation of the algorithm together with many examples and applications. The toric variety <em>X</em>, as well as the cohomology class <em>β</em>, must be defined using the package <span>Oscar.jl</span>.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"125 ","pages":"Article 102330"},"PeriodicalIF":0.7,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140650816","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 the computation of Gröbner bases for matrix-weighted homogeneous systems","authors":"Thibaut Verron","doi":"10.1016/j.jsc.2024.102327","DOIUrl":"https://doi.org/10.1016/j.jsc.2024.102327","url":null,"abstract":"<div><p>In this paper, we examine the structure of systems that are weighted homogeneous for several systems of weights, and how it impacts the computation of Gröbner bases. We present several linear algebra algorithms for computing Gröbner bases for systems with this structure, either directly or by reducing to existing structures. We also present suitable optimization techniques.</p><p>As an opening towards complexity studies, we discuss potential definitions of regularity and prove that they are generic if non-empty. Finally, we present experimental data from a prototype implementation of the algorithms in SageMath.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"125 ","pages":"Article 102327"},"PeriodicalIF":0.7,"publicationDate":"2024-04-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0747717124000312/pdfft?md5=9359f848e1aedeb40ee916bd1ff7229c&pid=1-s2.0-S0747717124000312-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140552713","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}