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Two-step Newton's method for deflation-one singular zeros of analytic systems 解压缩的两步牛顿法——解析系统的一个奇异零
IF 0.7 4区 数学
Journal of Symbolic Computation Pub Date : 2023-11-20 DOI: 10.1016/j.jsc.2023.102278
Kisun Lee , Nan Li , Lihong Zhi
{"title":"Two-step Newton's method for deflation-one singular zeros of analytic systems","authors":"Kisun Lee ,&nbsp;Nan Li ,&nbsp;Lihong Zhi","doi":"10.1016/j.jsc.2023.102278","DOIUrl":"https://doi.org/10.1016/j.jsc.2023.102278","url":null,"abstract":"<div><p><span>We propose a two-step Newton's method for refining an </span>approximation<span> of a singular zero whose deflation process terminates after one step, also known as a deflation-one singularity. Given an isolated singular zero of a square analytic system<span>, our algorithm exploits an invertible linear operator obtained by combining the Jacobian and a projection of the Hessian in the direction of the kernel of the Jacobian. We prove the quadratic convergence of the two-step Newton method when it is applied to an approximation of a deflation-one singular zero. Also, the algorithm requires a smaller size of matrices than the existing methods, making it more efficient. We demonstrate examples and experiments to show the efficiency of the method.</span></span></p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138448278","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
Toward finiteness of central configurations for the planar six-body problem by symbolic computations. (I) Determine diagrams and orders 用符号计算探讨平面六体问题中心构型的有限性。(1)确定图表和顺序
IF 0.7 4区 数学
Journal of Symbolic Computation Pub Date : 2023-11-17 DOI: 10.1016/j.jsc.2023.102277
Ke-Ming Chang, Kuo-Chang Chen
{"title":"Toward finiteness of central configurations for the planar six-body problem by symbolic computations. (I) Determine diagrams and orders","authors":"Ke-Ming Chang,&nbsp;Kuo-Chang Chen","doi":"10.1016/j.jsc.2023.102277","DOIUrl":"https://doi.org/10.1016/j.jsc.2023.102277","url":null,"abstract":"<div><p><span>In a series of papers we develop symbolic computation algorithms to investigate finiteness of central configurations for the planar </span><em>n</em>-body problem. Our approach is based on Albouy-Kaloshin's work on finiteness of central configurations for the 5-body problems. In their paper, bicolored graphs called <em>zw</em><span><span>-diagrams were introduced for possible scenarios when the finiteness conjecture fails, and proving finiteness amounts to exclusions of central configurations associated to these diagrams. Following their method, the amount of computations becomes enormous when there are more than five bodies. In this paper we introduce matrix algebra for determination of possible diagrams and asymptotic orders, devise several criteria to reduce </span>computational complexity, and determine possible </span><em>zw</em>-diagrams by automated deductions. For the planar six-body problem, we show that there are at most 86 <em>zw</em>-diagrams.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138453648","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
Explainable AI Insights for Symbolic Computation: A case study on selecting the variable ordering for cylindrical algebraic decomposition 符号计算中可解释的人工智能洞察:圆柱代数分解中变量排序选择的案例研究
IF 0.7 4区 数学
Journal of Symbolic Computation Pub Date : 2023-11-15 DOI: 10.1016/j.jsc.2023.102276
Lynn Pickering , Tereso del Río Almajano , Matthew England , Kelly Cohen
{"title":"Explainable AI Insights for Symbolic Computation: A case study on selecting the variable ordering for cylindrical algebraic decomposition","authors":"Lynn Pickering ,&nbsp;Tereso del Río Almajano ,&nbsp;Matthew England ,&nbsp;Kelly Cohen","doi":"10.1016/j.jsc.2023.102276","DOIUrl":"https://doi.org/10.1016/j.jsc.2023.102276","url":null,"abstract":"<div><p>In recent years there has been increased use of machine learning (ML) techniques within mathematics, including symbolic computation where it may be applied safely to optimise or select algorithms. This paper explores whether using explainable AI (XAI) techniques on such ML models can offer new insight for symbolic computation, inspiring new implementations within computer algebra systems that do not directly call upon AI tools. We present a case study on the use of ML to select the variable ordering for cylindrical algebraic decomposition. It has already been demonstrated that ML can make the choice well, but here we show how the SHAP tool for explainability can be used to inform new heuristics of a size and complexity similar to those human-designed heuristics currently commonly used in symbolic computation.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0747717123000901/pdfft?md5=ebb7a5437d38ce1efee92fa91cbda5ec&pid=1-s2.0-S0747717123000901-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138396787","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}
引用次数: 0
Syzygies, constant rank, and beyond Syzygies，固定等级，甚至更多
IF 0.7 4区 数学
Journal of Symbolic Computation Pub Date : 2023-11-14 DOI: 10.1016/j.jsc.2023.102274
Marc Härkönen , Lisa Nicklasson , Bogdan Raiţă
{"title":"Syzygies, constant rank, and beyond","authors":"Marc Härkönen ,&nbsp;Lisa Nicklasson ,&nbsp;Bogdan Raiţă","doi":"10.1016/j.jsc.2023.102274","DOIUrl":"10.1016/j.jsc.2023.102274","url":null,"abstract":"<div><p><span>We study linear PDE<span> with constant coefficients. The constant rank condition on a system of linear PDEs with constant coefficients is often used in the theory of compensated compactness. While this is a purely linear algebraic condition, the nonlinear algebra concept of </span></span>primary decomposition<span> is another important tool for studying such system of PDEs. In this paper we investigate the connection between these two concepts. From the nonlinear analysis<span> point of view, we make some progress in the study of weak lower semicontinuity of integral functionals defined on sequences of PDE constrained fields, when the PDEs do not have constant rank.</span></span></p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135716803","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: 3
Squarefree normal representation of zeros of zero-dimensional polynomial systems 零维多项式系统零的无平方正态表示
IF 0.7 4区 数学
Journal of Symbolic Computation Pub Date : 2023-11-14 DOI: 10.1016/j.jsc.2023.102273
Juan Xu , Dongming Wang , Dong Lu
{"title":"Squarefree normal representation of zeros of zero-dimensional polynomial systems","authors":"Juan Xu ,&nbsp;Dongming Wang ,&nbsp;Dong Lu","doi":"10.1016/j.jsc.2023.102273","DOIUrl":"10.1016/j.jsc.2023.102273","url":null,"abstract":"<div><p>For any zero-dimensional polynomial ideal <span><math><mi>I</mi></math></span> and any nonzero polynomial <em>F</em>, this paper shows that the union of the multi-set of zeros of the ideal sum <span><math><mi>I</mi><mo>+</mo><mo>〈</mo><mi>F</mi><mo>〉</mo></math></span> and that of the ideal quotient <span><math><mi>I</mi><mo>:</mo><mo>〈</mo><mi>F</mi><mo>〉</mo></math></span> is equal to the multi-set of zeros of <span><math><mi>I</mi></math></span>, where zeros are counted with multiplicities. Based on this zero relation and the computation of Gröbner bases, a complete multiplicity-preserved algorithm is proposed to decompose any zero-dimensional polynomial set into finitely many squarefree normal triangular sets, resulting in a squarefree normal representation for the zeros of the polynomial set. In the representation the multiplicities of the zeros of the triangular sets can be read out directly. Examples and experiments are presented to illustrate the algorithm and its performance.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135764366","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
Formations of finite groups in polynomial time: F-residuals and F-subnormality 多项式时间有限群的形成:f -残差和f -次正态
IF 0.7 4区 数学
Journal of Symbolic Computation Pub Date : 2023-09-25 DOI: 10.1016/j.jsc.2023.102271
Viachaslau I. Murashka
{"title":"Formations of finite groups in polynomial time: F-residuals and F-subnormality","authors":"Viachaslau I. Murashka","doi":"10.1016/j.jsc.2023.102271","DOIUrl":"https://doi.org/10.1016/j.jsc.2023.102271","url":null,"abstract":"<div><p>For a wide family of formations <span><math><mi>F</mi></math></span> it is proved that the <span><math><mi>F</mi></math></span><span>-residual of a permutation<span> finite group can be computed in polynomial time. Moreover, if in the previous case </span></span><span><math><mi>F</mi></math></span> is hereditary, then the <span><math><mi>F</mi></math></span>-subnormality of a subgroup can be checked in polynomial time.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49766599","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
Representation of non-special curves of genus 5 as plane sextic curves and its application to finding curves with many rational points 属5的非特殊曲线表示为平面六分曲线及其在求多有理点曲线中的应用
IF 0.7 4区 数学
Journal of Symbolic Computation Pub Date : 2023-09-25 DOI: 10.1016/j.jsc.2023.102272
Momonari Kudo , Shushi Harashita
{"title":"Representation of non-special curves of genus 5 as plane sextic curves and its application to finding curves with many rational points","authors":"Momonari Kudo ,&nbsp;Shushi Harashita","doi":"10.1016/j.jsc.2023.102272","DOIUrl":"https://doi.org/10.1016/j.jsc.2023.102272","url":null,"abstract":"<div><p>In algebraic geometry, it is important to provide effective parametrizations for families of curves, both in theory and in practice. In this paper, we present such an effective parametrization for the moduli of genus-5 curves that are neither hyperelliptic nor trigonal. Subsequently, we construct an algorithm for a complete enumeration of non-special genus-5 curves having more rational points than a specified bound, where “non-special curve” means that the curve is non-hyperelliptic and non-trigonal with mild singularities of the associated sextic model that we propose. As a practical application, we implement this algorithm using the computer algebra system MAGMA, specifically for curves over the prime field of characteristic 3.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49766602","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
Topological types of actions on curves 作用在曲线上的拓扑类型
IF 0.7 4区 数学
Journal of Symbolic Computation Pub Date : 2023-09-01 DOI: 10.1016/j.jsc.2023.01.002
Diego Conti , Alessandro Ghigi , Roberto Pignatelli
{"title":"Topological types of actions on curves","authors":"Diego Conti ,&nbsp;Alessandro Ghigi ,&nbsp;Roberto Pignatelli","doi":"10.1016/j.jsc.2023.01.002","DOIUrl":"https://doi.org/10.1016/j.jsc.2023.01.002","url":null,"abstract":"<div><p>We describe an algorithm that constructs a list of all topological types of holomorphic actions of a finite group on a compact Riemann surface <em>C</em> of genus <span><math><mi>g</mi><mo>≥</mo><mn>2</mn></math></span> with <span><math><mi>C</mi><mo>/</mo><mi>G</mi><mo>≅</mo><msup><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49760311","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: 3
Equivalence and reduction of bivariate polynomial matrices to their Smith forms 二元多项式矩阵的等价与约简
IF 0.7 4区 数学
Journal of Symbolic Computation Pub Date : 2023-09-01 DOI: 10.1016/j.jsc.2023.01.001
Dong Lu , Dingkang Wang , Fanghui Xiao , Xiaopeng Zheng
{"title":"Equivalence and reduction of bivariate polynomial matrices to their Smith forms","authors":"Dong Lu ,&nbsp;Dingkang Wang ,&nbsp;Fanghui Xiao ,&nbsp;Xiaopeng Zheng","doi":"10.1016/j.jsc.2023.01.001","DOIUrl":"https://doi.org/10.1016/j.jsc.2023.01.001","url":null,"abstract":"<div><p><span>This paper is concerned with Smith forms of bivariate<span> polynomial matrices. For a bivariate polynomial </span></span>square matrix<span> with the determinant being the product of two distinct and irreducible univariate polynomials<span>, we prove that it is equivalent to its Smith form. We design an algorithm to reduce this class of bivariate polynomial matrices to their Smith forms, and an example is given to illustrate the algorithm. Furthermore, we extend the above class of matrices to a more general case, and derive a necessary and sufficient condition for the equivalence of a matrix and one of its all possible existing Smith forms.</span></span></p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49754116","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: 1
Lower bounds on the rank and symmetric rank of real tensors 实张量的秩和对称秩的下界
IF 0.7 4区 数学
Journal of Symbolic Computation Pub Date : 2023-09-01 DOI: 10.1016/j.jsc.2023.01.004
Kexin Wang , Anna Seigal
{"title":"Lower bounds on the rank and symmetric rank of real tensors","authors":"Kexin Wang ,&nbsp;Anna Seigal","doi":"10.1016/j.jsc.2023.01.004","DOIUrl":"https://doi.org/10.1016/j.jsc.2023.01.004","url":null,"abstract":"<div><p><span>We lower bound the rank of a tensor by a linear combination of the ranks of three of its unfoldings, using Sylvester's rank inequality. In a similar way, we lower bound the symmetric rank by a linear combination of the symmetric ranks of three unfoldings. Lower bounds on the rank and symmetric rank of tensors are important for finding </span>counterexamples to Comon's conjecture. A real counterexample to Comon's conjecture is a tensor whose real rank and real symmetric rank differ. Previously, only one real counterexample was known, constructed in a paper of Shitov. We divide the construction into three steps. The first step involves linear spaces of binary tensors. The second step considers a linear space of larger decomposable tensors. The third step is to verify a conjecture that lower bounds the symmetric rank, on a tensor of interest. We use the construction to build an order six real tensor whose real rank and real symmetric rank differ.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49760255","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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