Journal of Symbolic Computation最新文献_第6页

A short proof for the parameter continuation theorem 参数延续定理的简短证明

IF 0.6 4区数学

Journal of Symbolic Computation Pub Date : 2024-08-03 DOI: 10.1016/j.jsc.2024.102373

Viktoriia Borovik, Paul Breiding

引用次数: 0

Fast evaluation and root finding for polynomials with floating-point coefficients 浮点系数多项式的快速求值和求根

IF 0.6 4区数学

Journal of Symbolic Computation Pub Date : 2024-07-20 DOI: 10.1016/j.jsc.2024.102372

Rémi Imbach , Guillaume Moroz

{"title":"Fast evaluation and root finding for polynomials with floating-point coefficients","authors":"Rémi Imbach , Guillaume Moroz","doi":"10.1016/j.jsc.2024.102372","DOIUrl":"10.1016/j.jsc.2024.102372","url":null,"abstract":"<div>Evaluating or finding the roots of a polynomial <math><mi>f</mi><mo>(</mo><mi>z</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>d</mi></mrow></msub><msup><mrow><mi>z</mi></mrow><mrow><mi>d</mi></mrow></msup></math> with floating-point number coefficients is a ubiquitous problem. By using a piecewise approximation of f obtained with a careful use of the Newton polygon of f, we improve state-of-the-art upper bounds on the number of operations to evaluate and find the roots of a polynomial. In particular, if the coefficients of f are given with m significant bits, we provide for the first time an algorithm that finds all the roots of f with a relative condition number lower than <math><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup></math>, using a number of bit operations quasi-linear in the bit-size of the floating-point representation of f. Notably, our new approach handles efficiently polynomials with coefficients ranging from <math><msup><mrow><mn>2</mn></mrow><mrow><mo>−</mo><mi>d</mi></mrow></msup></math> to <math><msup><mrow><mn>2</mn></mrow><mrow><mi>d</mi></mrow></msup></math>, both in theory and in practice.</div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102372"},"PeriodicalIF":0.6,"publicationDate":"2024-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141947541","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

Computing character tables and Cartan matrices of finite monoids with fixed point counting 用定点计数计算有限单体的字符表和卡坦矩阵

IF 0.6 4区数学

Journal of Symbolic Computation Pub Date : 2024-07-20 DOI: 10.1016/j.jsc.2024.102371

Balthazar Charles

引用次数: 0

Signature-based standard basis algorithm under the framework of GVW algorithm GVW 算法框架下基于签名的标准基础算法

IF 0.6 4区数学

Journal of Symbolic Computation Pub Date : 2024-07-19 DOI: 10.1016/j.jsc.2024.102370

Dong Lu , Dingkang Wang , Fanghui Xiao , Xiaopeng Zheng

引用次数: 0

On symmetry adapted bases in trigonometric optimization 关于三角优化中的对称适配基

IF 0.6 4区数学

Journal of Symbolic Computation Pub Date : 2024-07-17 DOI: 10.1016/j.jsc.2024.102369

Tobias Metzlaff

{"title":"On symmetry adapted bases in trigonometric optimization","authors":"Tobias Metzlaff","doi":"10.1016/j.jsc.2024.102369","DOIUrl":"10.1016/j.jsc.2024.102369","url":null,"abstract":"<div>The problem of computing the global minimum of a trigonometric polynomial is computationally hard. We address this problem for the case, where the polynomial is invariant under the exponential action of a finite group. The strategy is to follow an established relaxation strategy in order to obtain a converging hierarchy of lower bounds. Those bounds are obtained by numerically solving semi-definite programs (SDPs) on the cone of positive semi-definite Hermitian Toeplitz matrices, which is outlined in the book of Dumitrescu Dumitrescu (2007). To exploit the invariance, we show that the group has an induced action on the Toeplitz matrices and prove that the feasible region of the SDP can be restricted to the invariant matrices, whilst retaining the same solution. Then we construct a symmetry adapted basis tailored to this group action, which allows us to block-diagonalize invariant matrices and thus reduce the computational complexity to solve the SDP.The approach is in its generality novel for trigonometric optimization and complements the one that was proposed as a poster at the ISSAC 2022 conference Hubert et al. (2022) and later extended to Hubert et al. (2024). In the previous work, we first used the invariance of the trigonometric polynomial to obtain a classical polynomial optimization problem on the orbit space and subsequently relaxed the problem to an SDP. Now, we first make the relaxation and then exploit invariance.Partial results of this article have been presented as a poster at the ISSAC 2023 conference Metzlaff (2023).</div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102369"},"PeriodicalIF":0.6,"publicationDate":"2024-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0747717124000737/pdfft?md5=5897477ca2a3bd75103332cdce197f3a&pid=1-s2.0-S0747717124000737-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141779956","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

Approximate GCD of several multivariate sparse polynomials based on SLRA interpolation 基于 SLRA 插值的多个多元稀疏多项式的近似 GCD

IF 0.6 4区数学

Journal of Symbolic Computation Pub Date : 2024-07-15 DOI: 10.1016/j.jsc.2024.102368

Kosaku Nagasaka

{"title":"Approximate GCD of several multivariate sparse polynomials based on SLRA interpolation","authors":"Kosaku Nagasaka","doi":"10.1016/j.jsc.2024.102368","DOIUrl":"10.1016/j.jsc.2024.102368","url":null,"abstract":"<div>To compute the greatest common divisor (GCD) of a set of multivariate polynomials, modular algorithms are typically employed to prevent any growth in the coefficient polynomials in the intermediate expressions. However, when dealing with multivariate polynomials with a priori errors on their coefficients, using such modular algorithms becomes challenging. This is because any resulting approximate GCD computed in one variable may have perturbations depending on the evaluation point and may not be an image of the same desired multivariate approximate GCD. This necessitates computing it as given multivariate polynomials, and operating with large matrices whose size is exponential in the number of variables. In this paper, we present a new modular algorithm, suitable for dense cases and effective for sparse ones, called “SLRA interpolation”. This algorithm uses the multidimensional fast Fourier transform (FFT) and the structured low-rank approximation (SLRA) of non-square block diagonal matrices. The SLRA interpolation technique may reduce the time-complexity for one iteration in the computation of approximate GCD of several multivariate polynomials, especially for the sparse case.</div>","PeriodicalId":50031,"journal":{"name":"Journal of Symbolic Computation","volume":"127 ","pages":"Article 102368"},"PeriodicalIF":0.6,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141712328","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

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>Given two polynomials a and b in <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> which have no non-trivial common divisors, we prove that a generator of the elimination ideal <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> can be computed in quasi-linear time. To achieve this, we propose a randomized algorithm of the Monte Carlo type which requires <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> bit operations, where d and e bound the input degrees in x and in y respectively.The same complexity estimate applies to the computation of the largest degree invariant factor of the Sylvester matrix associated with a and b (with respect to either x or y), and of the resultant of a and b if they are sufficiently generic, in particular such that the Sylvester matrix has a unique non-trivial invariant factor.Our approach is to exploit reductions to problems of minimal polynomials in quotient algebras of the form <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>. 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.</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

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 , Marcelo Valdettaro","doi":"10.1016/j.jsc.2024.102359","DOIUrl":"10.1016/j.jsc.2024.102359","url":null,"abstract":"<div>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.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 f 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 f, or equivalently the maximum rank of the matrices in the Gram spectrahedron of f. 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 n squares and have common complex roots, and examples of polynomials in the boundary with SOS length larger than the expected one from the dimension.</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

引用次数: 0