{"title":"Reduction-based creative telescoping for definite summation of D-finite functions","authors":"Hadrien Brochet, 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}
{"title":"Positive definiteness of infinite and finite dimensional generalized Hilbert tensors and generalized Cauchy tensor","authors":"Yujin Paek, Jinhyok Kim, Songryong Pak","doi":"10.1016/j.jsc.2024.102326","DOIUrl":"https://doi.org/10.1016/j.jsc.2024.102326","url":null,"abstract":"<div><p>An Infinite and finite dimensional generalized Hilbert tensor with <em>a</em> is positive definite if and only if <span><math><mi>a</mi><mo>></mo><mn>0</mn></math></span>. The infinite dimensional generalized Hilbert tensor related operators <span><math><msub><mrow><mi>F</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> and <span><math><msub><mrow><mi>T</mi></mrow><mrow><mo>∞</mo></mrow></msub></math></span> are bounded, continuous and positively homogeneous. A generalized Cauchy tensor of which generating vectors are <span><math><mi>c</mi><mo>,</mo><mi>d</mi></math></span> is positive definite if and only if every element of vector <em>d</em> is not zero and each element of vector <em>c</em> is positive and mutually distinct. The 4th order <em>n</em>-dimensional generalized Cauchy tensor is matrix positive semi-definite if and only if every element of generating vector <em>c</em> is positive. Finally, the other properties of generalized Cauchy tensor are presented.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"125 ","pages":"Article 102326"},"PeriodicalIF":0.7,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140548504","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 parametric semidefinite programming with unknown boundaries","authors":"Jonathan D. Hauenstein , Tingting Tang","doi":"10.1016/j.jsc.2024.102324","DOIUrl":"https://doi.org/10.1016/j.jsc.2024.102324","url":null,"abstract":"<div><p>In this paper, we study parametric semidefinite programs (SDPs) where the solution space of both the primal and dual problems change simultaneously. Given a bounded set, we aim to find the <em>a priori</em> unknown maximal permissible perturbation set within it where the semidefinite program problem has a unique optimum and is analytic with respect to the parameters. Our approach reformulates the parametric SDP as a system of partial differential equations (PDEs) where this maximal analytical permissible set (MAPS) is the set on which the system of PDEs is well-posed. A sweeping Euler scheme is developed to approximate this <em>a priori</em> unknown perturbation set. We prove local and global error bounds for this second-order sweeping Euler scheme and demonstrate the method in comparison to existing SDP solvers and its performance on several two-parameter and three-parameter SDPs for which the MAPS can be visualized.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"125 ","pages":"Article 102324"},"PeriodicalIF":0.7,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140540318","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":"Short proofs of ideal membership","authors":"Clemens Hofstadler , Thibaut Verron","doi":"10.1016/j.jsc.2024.102325","DOIUrl":"https://doi.org/10.1016/j.jsc.2024.102325","url":null,"abstract":"<div><p>A cofactor representation of an ideal element, that is, a representation in terms of the generators, can be considered as a certificate for ideal membership. Such a representation is typically not unique, and some can be a lot more complicated than others. In this work, we consider the problem of computing sparsest cofactor representations, i.e., representations with a minimal number of terms, of a given element in a polynomial ideal. While we focus on the more general case of noncommutative polynomials, all results also apply to the commutative setting.</p><p>We show that the problem of computing cofactor representations with a bounded number of terms is decidable and <span><math><mtext>NP</mtext></math></span>-complete. Moreover, we provide a practical algorithm for computing sparse (not necessarily optimal) representations by translating the problem into a linear optimization problem and by exploiting properties of signature-based Gröbner basis algorithms. We show that, for a certain class of ideals, representations computed by this method are actually optimal, and we present experimental data illustrating that it can lead to noticeably sparser cofactor representations.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"125 ","pages":"Article 102325"},"PeriodicalIF":0.7,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0747717124000294/pdfft?md5=fc06471a76a7e331737ea355a494162b&pid=1-s2.0-S0747717124000294-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140540319","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":"On the connectedness of multistationarity regions of small reaction networks","authors":"Allison McClure, Anne Shiu","doi":"10.1016/j.jsc.2024.102323","DOIUrl":"10.1016/j.jsc.2024.102323","url":null,"abstract":"<div><p>A multistationarity region is the part of a reaction network's parameter space that gives rise to multiple steady states. Mathematically, this region consists of the positive parameters for which a parametrized family of polynomial equations admits two or more positive roots. Much recent work has focused on analyzing multistationarity regions of biologically significant reaction networks and determining whether such regions are connected; indeed, a better understanding of the topology and geometry of such regions may help elucidate how robust multistationarity is to perturbations. Here we focus on the multistationarity regions of small networks, those with few species and few reactions. For two families of such networks – those with one species and up to three reactions, and those with two species and up to two reactions – we prove that the resulting multistationarity regions are connected. We also give an example of a network with one species and six reactions for which the multistationarity region is disconnected. Our proofs rely on the formula for the discriminant of a trinomial, a classification of small multistationary networks, and a recent result of Feliu and Telek that partially generalizes Descartes' rule of signs.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"125 ","pages":"Article 102323"},"PeriodicalIF":0.7,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140196310","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 integral closure of a primary ideal is not always primary","authors":"Nan Li , Zijia Li , Zhi-Hong Yang , Lihong Zhi","doi":"10.1016/j.jsc.2024.102315","DOIUrl":"10.1016/j.jsc.2024.102315","url":null,"abstract":"<div><p>In <span>1936</span>, Krull asked if the integral closure of a primary ideal is still primary. Fifty years later, Huneke partially answered this question by giving a primary polynomial ideal whose integral closure is not primary in a regular local ring of characteristic <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span>. We provide counterexamples to Krull's question regarding polynomial rings over any fields. We also find that the Jacobian ideal <em>J</em> of the polynomial <span><math><mi>f</mi><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>6</mn></mrow></msup><mo>+</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>6</mn></mrow></msup><mo>+</mo><msup><mrow><mi>x</mi></mrow><mrow><mn>4</mn></mrow></msup><mi>z</mi><mi>t</mi><mo>+</mo><msup><mrow><mi>z</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> given by <span>Briançon and Speder (1975)</span> is a counterexample to Krull's question.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"125 ","pages":"Article 102315"},"PeriodicalIF":0.7,"publicationDate":"2024-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140071563","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":"Solving polynomial systems over non-fields and applications to modular polynomial factoring","authors":"Sayak Chakrabarti, Ashish Dwivedi, Nitin Saxena","doi":"10.1016/j.jsc.2024.102314","DOIUrl":"10.1016/j.jsc.2024.102314","url":null,"abstract":"<div><p>We study the problem of solving a system of <em>m</em> polynomials in <em>n</em> variables over the ring of integers modulo a prime-power <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span>. The problem over finite fields is well studied in varied parameter settings. For small characteristic <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span>, Lokshtanov et al. (SODA'17) initiated the study, for degree <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span> systems, to improve the exhaustive search complexity of <span><math><mi>O</mi><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo><mo>⋅</mo><mtext>poly</mtext><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> to <span><math><mi>O</mi><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>0.8765</mn><mi>n</mi></mrow></msup><mo>)</mo><mo>⋅</mo><mtext>poly</mtext><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span>; which currently is improved to <span><math><mi>O</mi><mo>(</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>0.6943</mn><mi>n</mi></mrow></msup><mo>)</mo><mo>⋅</mo><mtext>poly</mtext><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> in Dinur (SODA'21). For large <em>p</em> but constant <em>n</em>, Huang and Wong (FOCS'96) gave a randomized <span><math><mtext>poly</mtext><mo>(</mo><mi>d</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>log</mi><mo></mo><mi>p</mi><mo>)</mo></math></span> time algorithm. Note that for growing <em>n</em>, system-solving is known to be <em>intractable</em> even with <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span> and degree <span><math><mi>d</mi><mo>=</mo><mn>2</mn></math></span>.</p><p>We devise a randomized <span><math><mtext>poly</mtext><mo>(</mo><mi>d</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>log</mi><mo></mo><mi>p</mi><mo>)</mo></math></span>-time algorithm to find a root of a given system of <em>m</em> integral polynomials of degrees bounded by <em>d</em>, in <em>n</em> variables, modulo a prime power <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span>; when <span><math><mi>n</mi><mo>+</mo><mi>k</mi></math></span> is constant. In a way, we extend the efficient algorithm of Huang and Wong (FOCS'96) for system-solving over Galois fields (i.e., characteristic <em>p</em>) to system-solving over Galois <em>rings</em> (i.e., characteristic <span><math><msup><mrow><mi>p</mi></mrow><mrow><mi>k</mi></mrow></msup></math></span>); when <span><math><mi>k</mi><mo>></mo><mn>1</mn></math></span> is constant. The challenge here is to find a lift of <em>singular</em> <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-roots (exponentially many); as there is no efficient general way known in algebraic-geometry for resolving singularities.</p><p>Our algorithm has applications to factoring univariate polynomials over Galois rings. Given <span><math><mi>f</mi><mo>∈</mo><mi>Z</mi><mo>[</mo><mi>x</mi><mo>]","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"125 ","pages":"Article 102314"},"PeriodicalIF":0.7,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140071367","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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