Ludovic Brieulle, L. Feo, Javad Doliskani, J. Flori, É. Schost
{"title":"Computing isomorphisms and embeddings of finite fields","authors":"Ludovic Brieulle, L. Feo, Javad Doliskani, J. Flori, É. Schost","doi":"10.1090/mcom/3363","DOIUrl":"https://doi.org/10.1090/mcom/3363","url":null,"abstract":"Let <i>q</i> be a prime power and let F<sub><i>q</i></sub> be a field with <i>q</i> elements. Let <i>f</i> and <i>g</i> be irreducible polynomials in F<sub><i>q</i></sub>[<i>X</i>], with deg <i>f</i> dividing deg <i>g.</i> Define <i>k</i> = F<sub><i>q</i></sub>[<i>X</i>]/<i>f</i> and <i>K</i> = F<sub><i>q</i></sub>[<i>X</i>]/<i>g</i>, then there is an embedding <i>φ</i> : <i>k</i> [EQUATION] <i>K</i>, unique up to F<sub><i>q</i></sub>-automorphisms of <i>k.</i> Our goal is to describe algorithms to efficiently represent and evaluate one such embedding.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"12 1","pages":"117-119"},"PeriodicalIF":0.0,"publicationDate":"2017-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87829909","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The bohemian eigenvalue project","authors":"Robert M Corless, Steven E. Thornton","doi":"10.1145/3055282.3055289","DOIUrl":"https://doi.org/10.1145/3055282.3055289","url":null,"abstract":"Bohemian eigenvalues are the eigenvalues of matrices with entries of bounded height, typically drawn from a discrete set. We will call this set F with cardinality #F. The name \"Bohemian\" is intended as a mnemonic and is derived from \"bounded height integer matrices.\" These objects are surprisingly interesting to study, with many unsolved problems related to them, and with many applications. See the works of Tao and Vu [10] for universality results for larger dimension in the generic structured case, for instance. This project concentrates on explicit construction of high resolution pictures of the eigenvalues for modest dimensions and sizes of the entries; for instance, Figure 1a is a picture of the eigenvalues of all 5 × 5 matrices with entries in {−1, 0, 1} colored by density and plotted on the complex plane.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"1 1","pages":"158-160"},"PeriodicalIF":0.0,"publicationDate":"2017-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89733411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Denominator bounds for higher order systems of linear recurrence equations","authors":"J. Middeke, Carsten Schneider","doi":"10.1145/3055282.3055298","DOIUrl":"https://doi.org/10.1145/3055282.3055298","url":null,"abstract":"Let (<i>K</i>, <i>σ</i>) be a difference field. We define the set of constants by const <i>K</i> = {<i>c</i> ∈ <i>K</i> | <i>σ</i>(<i>c</i>) = <i>c</i>}. A ΠΣ*-<i>extension</i> of <i>K</i> is a field of rational functions <i>K</i>(<i>t</i>) over <i>K</i> together with an extension of <i>σ</i> to <i>K</i>(<i>t</i>) given by either <i>σ</i>(<i>t</i>) = <i>at</i> (Π case) or <i>σ</i>(<i>t</i>) = <i>t</i> + <i>b</i> (Σ* case) for some non-zero <i>a</i> or <i>b</i> ∈ <i>K</i> such that const <i>K</i>(<i>t</i>) = const <i>K</i> holds. See, for example, [8, 9] for more details on ΠΣ*-extensions.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"48 1","pages":"185-187"},"PeriodicalIF":0.0,"publicationDate":"2017-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90644895","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Real limit points of quasi-componenets of regular chains","authors":"P. Alvandi, M. M. Maza","doi":"10.1145/3055282.3055286","DOIUrl":"https://doi.org/10.1145/3055282.3055286","url":null,"abstract":"The work reported here is motivated by problems arising in solving polynomial systems over the real numbers.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"1 1","pages":"148-150"},"PeriodicalIF":0.0,"publicationDate":"2017-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76851470","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A new idea on the interval-symbol method with correct zero rewriting for reducing exact computations","authors":"Akiyuki Katayama, Kiyoshi Shirayanagi","doi":"10.1145/3055282.3055295","DOIUrl":"https://doi.org/10.1145/3055282.3055295","url":null,"abstract":"The ISCZ method (Interval-Symbol method with Correct Zero rewriting) was proposed in [2] based on Shirayanagi-Sweedler stabilization theory ([3]), to reduce the amount of exact computations as much as possible to obtain the exact results. The authors of [2] applied this method to Buchberger's algorithm which computes a Gröbner basis, but the effectiveness was not achieved except for a few examples. This is not only because the complex structure of Buchberger's algorithm causes symbols to significantly grow but also because we naively implemented the ISCZ method without any particular devices. In this poster, we propose a new idea for efficiency of the ISCZ method and show its effect by applying it to calculation of Frobenius canonical form of square matrices. Jordan canonical form is also well-known, but it requires an extension of the field containing the roots of its characteristic polynomial. On the other hand, Frobenius canonical form can be computed by using only basic arithmetic operations, but nevertheless has almost the same information as Jordan canonical form.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"5 1","pages":"176-178"},"PeriodicalIF":0.0,"publicationDate":"2017-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77456909","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Representation of hypergeometric products in difference rings","authors":"E. Ocansey, Carsten Schneider","doi":"10.1145/3055282.3055290","DOIUrl":"https://doi.org/10.1145/3055282.3055290","url":null,"abstract":"In his pioneering work [1, 2], Michael Karr introduced ΠΣ-fields which provide a rather general framework for symbolic summation. He worked out the first algorithmic steps to represent indefinite nested sums and products as transcendental extensions over a computable ground field K called the field of constants. Furthermore, he presented an algorithm that solves the parameterized telescoping problem, and as special cases the telescoping and creative telescoping problems [3] within a given ΠΣ-field.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"83 1","pages":"161-163"},"PeriodicalIF":0.0,"publicationDate":"2017-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83975544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Fibonacci-mandelbrot polynomials and matrices","authors":"Eunice Y. S. Chan, Robert M Corless","doi":"10.1145/3055282.3055288","DOIUrl":"https://doi.org/10.1145/3055282.3055288","url":null,"abstract":"We explore a family of polynomials similar to the Mandelbrot polynomials called the Fibonacci-Mandelbrot polynomials defined by <i>q</i><sub>0</sub>(<i>z</i>) = 0, <i>q</i><sub>1</sub>(<i>z</i>) = 1, and <i>q<sub>n</sub></i>(<i>z</i>) = <i>zq</i><sub><i>n</i>−1</sub><i>q</i><sub><i>n</i>−2</sub> + 1. We compute the roots of the Fibonacci-Mandelbrot polynomials using two methods. One method uses a recursively constructed matrix, where elements are 0, 1, or −1, whose eigenvalues are the roots of <i>q<sub>n</sub></i>(<i>z</i>). The other method uses a special-purpose homotopy continuation method, where the solution of the differential equation, [EQUATION], in which the initial condition are 0, and the roots of <i>q</i><sub><i>n</i>−1</sub> and <i>q</i><sub><i>n</i>−2</sub>, are also the roots of the Fibonacci-Mandelbrot polynomials.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"92 1","pages":"155-157"},"PeriodicalIF":0.0,"publicationDate":"2017-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78204785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Z2-equivariant standard bases for submodules associated with Z2-equivariant singularities","authors":"M. Gazor, Mahsa Kazemi","doi":"10.1145/3055282.3055293","DOIUrl":"https://doi.org/10.1145/3055282.3055293","url":null,"abstract":"Let <i>x</i> = (<i>x</i><sub>1</sub>,...,<i>x</i><sub><i>n</i></sub>) ∈ R <sup><i>n</i></sup> and λ ∈ R. A smooth map <i>f</i>(<i>x</i>,λ) is called <i>Z</i><sub>2</sub>-equivariant (<i>Z</i><sub>2</sub>-invariant) if <i>f</i>(−<i>x</i>, λ) = −<i>f</i>(<i>x</i>,λ) (<i>f</i>(−<i>x</i>, λ) = <i>f</i>(<i>x</i>,λ)). Consider the local solutions of a <i>Z</i><sub>2</sub>-equivariant map <i>f</i>(<i>x</i>,λ) = 0 around a solution, say <i>f</i>(<i>x</i><sub>0</sub>,λ<sub>0</sub>), as the parameters smoothly vary. The solution set may experience a surprising behavior like observing changes in the number of solutions. Each of such problems/changes is called a singularity/bifurcation. Since our analysis is about local solutions, we call any two smooth map <i>f</i>(<i>x</i>,λ) and <i>g</i>(<i>x</i>,λ) as germ-equivalent when they are identical on a neighborhood of (<i>x</i><sub>0</sub>,λ<sub>0</sub>) = (0,0). Each germ equivalent class is referred to a smooth germ. The space of all smooth <i>Z</i><sub>2</sub>-equivariant germs is denoted by [EQUATION] and space of all smooth <i>Z</i><sub>2</sub>-invariant germs is denoted by [EQUATION]<sub><i>x</i>,λ</sub>(<i>Z</i><sub>2</sub>). The space [EQUATION] is a module over the ring of <i>Z</i><sub>2</sub>-invariant germs [EQUATION]<sub><i>x</i>,λ</sub>(<i>Z</i><sub>2</sub>); see [3, 2, 7] for more information and the origins of our notations.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"1 1","pages":"170-172"},"PeriodicalIF":0.0,"publicationDate":"2017-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88910624","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Efficient detection of hessian matrix sparsity pattern","authors":"R. Carter, S. Hossain, M. Sultana","doi":"10.1145/3055282.3055287","DOIUrl":"https://doi.org/10.1145/3055282.3055287","url":null,"abstract":"Evaluation of the Hessian matrix of a scalar function is a subproblem in many numerical optimization algorithms. For large-scale problems often the Hessian matrix is sparse and structured, and it is preferable to exploit such information when available. Using symmetry in the second derivative values of the components it is possible to detect the sparsity pattern of the Hessian via products of the Hessian matrix with specially chosen direction vectors. We use graph coloring methods and employ efficient sparse data structures to implement the sparsity pattern detection algorithms. Results from preliminary numerical testings are highly promising.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"23 1","pages":"151-154"},"PeriodicalIF":0.0,"publicationDate":"2017-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86117958","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Proof of a series solution for euler's trinomial equation","authors":"Fei Wang","doi":"10.1145/3055282.3055284","DOIUrl":"https://doi.org/10.1145/3055282.3055284","url":null,"abstract":"In 1779, Leonhard Euler published a paper about Lambert's transcendental equation in the symmetric form <i>x</i><sup><i>α</i></sup> − <i>x</i><sup><i>β</i></sup> = (<i>α</i> − <i>β</i>)<i>vx</i><sup><i>α</i>+<i>β</i></sup>. In the paper, he studied the series solution of this equation and other results based on an assumption which was not proved in the paper. Euler's paper gave the first series expanion for the so-called Lambert W function. In this work, we briefly review Euler's results and give a proof to modern standards of rigor of the series solution of Lambert's transcendental equation.","PeriodicalId":7093,"journal":{"name":"ACM Commun. Comput. Algebra","volume":"1 1","pages":"136-144"},"PeriodicalIF":0.0,"publicationDate":"2017-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83930373","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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