Ludovic Brieulle, L. Feo, Javad Doliskani, J. Flori, É. Schost
Let q be a prime power and let Fq be a field with q elements. Let f and g be irreducible polynomials in Fq[X], with deg f dividing deg g. Define k = Fq[X]/f and K = Fq[X]/g, then there is an embedding φ : k [EQUATION] K, unique up to Fq-automorphisms of k. Our goal is to describe algorithms to efficiently represent and evaluate one such embedding.
{"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}
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.
{"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}
Let (K, σ) be a difference field. We define the set of constants by const K = {c ∈ K | σ(c) = c}. A ΠΣ*-extension of K is a field of rational functions K(t) over K together with an extension of σ to K(t) given by either σ(t) = at (Π case) or σ(t) = t + b (Σ* case) for some non-zero a or b ∈ K such that const K(t) = const K holds. See, for example, [8, 9] for more details on ΠΣ*-extensions.
设(K， σ)是一个差分场。我们用const K = {c∈K | σ(c) = c}来定义常数集。K的ΠΣ*-扩展是有理函数K(t) / K的域以及σ到K(t)的扩展，由σ(t) = at (Π情况)或σ(t) = t + b (Σ*情况)给出，对于某些非零A或b∈K使得const K(t) = const K成立。例如，请参阅[8,9]，了解ΠΣ*-extensions的更多详细信息。
{"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}
The work reported here is motivated by problems arising in solving polynomial systems over the real numbers.
这里报告的工作是由解决实数上的多项式系统所产生的问题所激发的。
{"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}
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.
在Shirayanagi-Sweedler镇定理论([3])的基础上，[2]提出了ISCZ方法(Interval-Symbol method with Correct Zero rewrite)，以尽可能减少精确计算量以获得精确结果。[2]的作者将该方法应用于计算Gröbner基的Buchberger算法，但除了少数例子外，并没有达到有效性。这不仅是因为Buchberger算法的复杂结构导致符号显着增长，而且还因为我们在没有任何特定设备的情况下天真地实现了ISCZ方法。在这张海报中，我们提出了ISCZ方法效率的新思路，并通过将其应用于方阵的Frobenius标准形式的计算来展示其效果。Jordan标准形式也是众所周知的，但它需要一个包含其特征多项式根的域的扩展。另一方面，Frobenius标准形式可以通过基本的算术运算来计算，但与Jordan标准形式具有几乎相同的信息。
{"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}
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.
Michael Karr在他的开创性工作[1,2]中引入了ΠΣ-fields，它为符号求和提供了一个相当一般的框架。他提出了第一个算法步骤，将无限嵌套和和和表示为可计算的地面域K的超越扩展，称为常数域。此外，他还提出了一种算法来解决参数化伸缩问题，并作为特殊情况解决给定ΠΣ-field内的伸缩和创造性伸缩问题[3]。
{"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}
We explore a family of polynomials similar to the Mandelbrot polynomials called the Fibonacci-Mandelbrot polynomials defined by q0(z) = 0, q1(z) = 1, and qn(z) = zqn−1qn−2 + 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 qn(z). 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 qn−1 and qn−2, are also the roots of the Fibonacci-Mandelbrot polynomials.
{"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}
Let x = (x1,...,xn) ∈ R n and λ ∈ R. A smooth map f(x,λ) is called Z2-equivariant (Z2-invariant) if f(−x, λ) = −f(x,λ) (f(−x, λ) = f(x,λ)). Consider the local solutions of a Z2-equivariant map f(x,λ) = 0 around a solution, say f(x0,λ0), 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 f(x,λ) and g(x,λ) as germ-equivalent when they are identical on a neighborhood of (x0,λ0) = (0,0). Each germ equivalent class is referred to a smooth germ. The space of all smooth Z2-equivariant germs is denoted by [EQUATION] and space of all smooth Z2-invariant germs is denoted by [EQUATION]x,λ(Z2). The space [EQUATION] is a module over the ring of Z2-invariant germs [EQUATION]x,λ(Z2); see [3, 2, 7] for more information and the origins of our notations.
{"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}
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.
{"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}
In 1779, Leonhard Euler published a paper about Lambert's transcendental equation in the symmetric form xα − xβ = (α − β)vxα+β. 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.
{"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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