Journal of Symbolic Computation最新文献

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Bivariate polynomial reduction and elimination ideal over finite fields 有限域上的二元多项式还原和消元理想
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2024-07-15 DOI: 10.1016/j.jsc.2024.102367
Gilles Villard
{"title":"Bivariate polynomial reduction and elimination ideal over finite fields","authors":"Gilles Villard","doi":"10.1016/j.jsc.2024.102367","DOIUrl":"10.1016/j.jsc.2024.102367","url":null,"abstract":"<div><p>Given two polynomials <em>a</em> and <em>b</em> in <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>]</mo></math></span> which have no non-trivial common divisors, we prove that a generator of the elimination ideal <span><math><mo>〈</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>〉</mo><mo>∩</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>]</mo></math></span> can be computed in quasi-linear time. To achieve this, we propose a randomized algorithm of the Monte Carlo type which requires <span><math><msup><mrow><mo>(</mo><mi>d</mi><mi>e</mi><mi>log</mi><mo>⁡</mo><mi>q</mi><mo>)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span> bit operations, where <em>d</em> and <em>e</em> bound the input degrees in <em>x</em> and in <em>y</em> respectively.</p><p>The same complexity estimate applies to the computation of the largest degree invariant factor of the Sylvester matrix associated with <em>a</em> and <em>b</em> (with respect to either <em>x</em> or <em>y</em>), and of the resultant of <em>a</em> and <em>b</em> if they are sufficiently generic, in particular such that the Sylvester matrix has a unique non-trivial invariant factor.</p><p>Our approach is to exploit reductions to problems of minimal polynomials in quotient algebras of the form <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>]</mo><mo>/</mo><mo>〈</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>〉</mo></math></span>. By proposing a new method based on structured polynomial matrix division for computing with the elements of the quotient, we succeed in improving the best-known complexity bounds.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102367"},"PeriodicalIF":0.6,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141710357","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
Corrigendum to “On the cactus rank of cubics forms” [J. Symb. Comput. 50 (2013) 291–297] 关于立方体形式的仙人掌秩》的更正 [J. Symb.
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2024-07-15 DOI: 10.1016/j.jsc.2024.102354
Alessandra Bernardi , Kristian Ranestad
{"title":"Corrigendum to “On the cactus rank of cubics forms” [J. Symb. Comput. 50 (2013) 291–297]","authors":"Alessandra Bernardi ,&nbsp;Kristian Ranestad","doi":"10.1016/j.jsc.2024.102354","DOIUrl":"10.1016/j.jsc.2024.102354","url":null,"abstract":"","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102354"},"PeriodicalIF":0.6,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0747717124000580/pdfft?md5=cbba526584c007241f6c87b1cbfb6282&pid=1-s2.0-S0747717124000580-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141623268","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
Strictly positive polynomials in the boundary of the SOS cone SOS 锥体边界上的严格正多项式
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2024-07-11 DOI: 10.1016/j.jsc.2024.102359
Santiago Laplagne , Marcelo Valdettaro
{"title":"Strictly positive polynomials in the boundary of the SOS cone","authors":"Santiago Laplagne ,&nbsp;Marcelo Valdettaro","doi":"10.1016/j.jsc.2024.102359","DOIUrl":"10.1016/j.jsc.2024.102359","url":null,"abstract":"<div><p>We study the boundary of the cone of real polynomials that can be decomposed as a sum of squares (SOS) of real polynomials. This cone is included in the cone of nonnegative polynomials and both cones share a part of their boundary, which corresponds to polynomials that vanish at at least one point. We focus on the part of the boundary which is not shared, corresponding to strictly positive polynomials.</p><p>For the cases of polynomials of degree 6 in 3 variables and degree 4 in 4 variables, this boundary has been completely characterized by G. Blekherman. For the cases of a polynomial <em>f</em> in more variables or of higher degree, results by G. Blekherman, R. Sinn and M. Velasco and other authors based on a conjecture by Ottaviani and Paoletti give bounds for the maximum number of linearly independent polynomials that can appear in an SOS decomposition of <em>f</em>, or equivalently the maximum rank of the matrices in the Gram spectrahedron of <em>f</em>. We show that the same bounds can be obtained from the Eisenbud-Green-Harris conjecture. Combining theoretical results and computational techniques, we compute examples that allow us to prove the optimality of the bounds for all degrees and number of variables. Additionally, we give examples for the following problems: examples in the boundary of the cone that are the sum of less than <em>n</em> squares and have common complex roots, and examples of polynomials in the boundary with SOS length larger than the expected one from the dimension.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102359"},"PeriodicalIF":0.6,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718944","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
A semi-numerical algorithm for the homology lattice and periods of complex elliptic surfaces over P1 P1< 上复椭圆曲面的同调晶格和周期的半数值算法
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2024-07-08 DOI: 10.1016/j.jsc.2024.102357
Eric Pichon-Pharabod
{"title":"A semi-numerical algorithm for the homology lattice and periods of complex elliptic surfaces over P1","authors":"Eric Pichon-Pharabod","doi":"10.1016/j.jsc.2024.102357","DOIUrl":"10.1016/j.jsc.2024.102357","url":null,"abstract":"<div><p>We provide an algorithm for computing a basis of homology of elliptic surfaces over <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msubsup></math></span> that is sufficiently explicit for integration of periods to be carried out. This allows the heuristic recovery of several algebraic invariants of the surface, notably the Néron–Severi lattice, the transcendental lattice, the Mordell–Weil group and the Mordell–Weil lattice. This algorithm comes with a SageMath implementation.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102357"},"PeriodicalIF":0.6,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141692668","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
Dissimilar subalgebras of symmetry algebra of plasticity equations 塑性方程对称代数的异类子代数
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2024-07-06 DOI: 10.1016/j.jsc.2024.102358
Sergey I. Senashov , Alexander Yakhno
{"title":"Dissimilar subalgebras of symmetry algebra of plasticity equations","authors":"Sergey I. Senashov ,&nbsp;Alexander Yakhno","doi":"10.1016/j.jsc.2024.102358","DOIUrl":"10.1016/j.jsc.2024.102358","url":null,"abstract":"<div><p>In this paper we construct the optimal sets of dissimilar subalgebras up to dimension three for the Lie algebra of point symmetries of the system of three-dimensional stationary equations of perfect plasticity with the Huber–von Mises yield condition. The obtained results can be used to solve the problem of determining all invariant solutions of this system. It was necessary to design algorithms to facilitate some steps of the classification of subalgebras. The computational algebraic system SageMath was chosen to implement these algorithms. The most used functions and procedures are listed. The developed algorithms can be adapted to classify subalgebras of higher dimensions.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102358"},"PeriodicalIF":0.6,"publicationDate":"2024-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141630273","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
Determinant evaluations inspired by Di Francesco's determinant for twenty-vertex configurations 受迪弗朗西斯科行列式启发的二十顶点配置行列式评估
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2024-07-03 DOI: 10.1016/j.jsc.2024.102352
{"title":"Determinant evaluations inspired by Di Francesco's determinant for twenty-vertex configurations","authors":"","doi":"10.1016/j.jsc.2024.102352","DOIUrl":"10.1016/j.jsc.2024.102352","url":null,"abstract":"<div><p>In his work on the twenty vertex model, <span><span>Di Francesco (2021)</span></span> found a determinant formula for the number of configurations in a specific such model, and he conjectured a closed form product formula for the evaluation of this determinant. We prove this conjecture here. Moreover, we actually generalize this determinant evaluation to a one-parameter family of determinant evaluations, and we present many more determinant evaluations of similar type — some proved, some left open as conjectures.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102352"},"PeriodicalIF":0.6,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141574411","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
Graceful bases in solution spaces of differential and difference equations 微分方程和差分方程解空间中的优美基点
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2024-07-03 DOI: 10.1016/j.jsc.2024.102355
{"title":"Graceful bases in solution spaces of differential and difference equations","authors":"","doi":"10.1016/j.jsc.2024.102355","DOIUrl":"10.1016/j.jsc.2024.102355","url":null,"abstract":"<div><p>We construct fundamental systems of solutions to linear ordinary differential equations, linear difference equations, and systems of partial differential equations whose elements remain linearly independent for all values of algebraically independent symbolic parameters.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102355"},"PeriodicalIF":0.6,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141574413","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
On the log-concavity of the n-th root of sequences 论序列 n 次根的对数凹性
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2024-06-27 DOI: 10.1016/j.jsc.2024.102349
Ernest X.W. Xia , Zuo-Ru Zhang
{"title":"On the log-concavity of the n-th root of sequences","authors":"Ernest X.W. Xia ,&nbsp;Zuo-Ru Zhang","doi":"10.1016/j.jsc.2024.102349","DOIUrl":"10.1016/j.jsc.2024.102349","url":null,"abstract":"&lt;div&gt;&lt;p&gt;In recent years, the log-concavity of the &lt;em&gt;n&lt;/em&gt;-th root of a sequence &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mroot&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/mroot&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; has been received a lot of attention. Recently, Sun posed the following conjecture in his new book: the sequences &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mroot&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/mroot&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; and &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mroot&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/mroot&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; are log-concave, where&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;munderover&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; and&lt;span&gt;&lt;span&gt;&lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;:&lt;/mo&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;munderover&gt;&lt;mo&gt;∑&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/munderover&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;.&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; In this paper, two methods, semi-automatic and analytic methods, are used to confirm Sun's conjecture. The semi-automatic method relies on a criterion on the log-concavity of &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mroot&gt;&lt;mrow&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;S&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/mroot&gt;&lt;mo&gt;}&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt; given b","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102349"},"PeriodicalIF":0.6,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503505","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
On the dimension of the solution space of linear difference equations over the ring of infinite sequences 论无穷序列环上线性差分方程解空间的维度
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2024-06-27 DOI: 10.1016/j.jsc.2024.102350
Sergei Abramov , Gleb Pogudin
{"title":"On the dimension of the solution space of linear difference equations over the ring of infinite sequences","authors":"Sergei Abramov ,&nbsp;Gleb Pogudin","doi":"10.1016/j.jsc.2024.102350","DOIUrl":"10.1016/j.jsc.2024.102350","url":null,"abstract":"<div><p>For a linear difference equation with the coefficients being computable sequences, we establish algorithmic undecidability of the problem of determining the dimension of the solution space including the case when some additional prior information on the dimension is available.</p></div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102350"},"PeriodicalIF":0.6,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529011","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
Orthogonal-symplectic matrices and their parametric representation 正交-交错矩阵及其参数表示法
IF 0.6 4区 数学
Journal of Symbolic Computation Pub Date : 2024-06-27 DOI: 10.1016/j.jsc.2024.102353
Alexander Batkhin , Alexander Petrov
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