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On the units generated by Weierstrass forms 关于韦尔斯特拉斯表格生成的单位
Lms Journal of Computation and Mathematics Pub Date : 2014-01-01 DOI: 10.1112/S1461157014000163
Ömer Küçüksakalli
{"title":"On the units generated by Weierstrass forms","authors":"Ömer Küçüksakalli","doi":"10.1112/S1461157014000163","DOIUrl":"https://doi.org/10.1112/S1461157014000163","url":null,"abstract":"There is an algorithm of Schoof for finding divisors of class numbers of real cyclotomic fields of prime conductor. In this paper we introduce an improvement of the elliptic analogue of this algorithm by using a subgroup of elliptic units given by Weierstrass forms. These elliptic units which can be expressed in terms of $def xmlpi #1{}def mathsfbi #1{boldsymbol {mathsf {#1}}}let le =leqslant let leq =leqslant let ge =geqslant let geq =geqslant def Pr {mathit {Pr}}def Fr {mathit {Fr}}def Rey {mathit {Re}}x$ -coordinates of points on elliptic curves enable us to use the fast arithmetic of elliptic curves over finite fields.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"303-313"},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000163","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411323","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}
引用次数: 1
On the beta expansion of Salem numbers of degree 8 关于8度塞勒姆数的展开
Lms Journal of Computation and Mathematics Pub Date : 2014-01-01 DOI: 10.1112/S1461157014000035
Hachem Hichri
{"title":"On the beta expansion of Salem numbers of degree 8","authors":"Hachem Hichri","doi":"10.1112/S1461157014000035","DOIUrl":"https://doi.org/10.1112/S1461157014000035","url":null,"abstract":"Abstract Boyd showed that the beta expansion of Salem numbers of degree 4 were always eventuallyperiodic. Based on an heuristic argument, Boyd had conjectured that the same is true for Salemnumbers of degree 6 but not for Salem numbers of degree 8. This paper examines Salem numbersof degree 8 and collects experimental evidence in support of Boyd’s conjecture. 1. Introduction and basic de nitionsThe representations of numbers in a non-integer base >1 was pioneered by R\u0013enyi [11], wherehe introduced the beta expansion (called also greedy expansion) to represent any real numberxof the interval [0;1] in base by a sequence of digits x 1 x 2 x 3 :::which can be computed bythe following algorithm.Greedy algorithm. Denote by bycand fygthe integer part and the fractional part of areal number y, respectively.Set r 0 = xand for i> 1, x i = b r i 1 c, r i = f r i 1 g.Or, similarly, using the beta transformation T= T of the unit interval which is the mapping:T: [0;1] ! [0;1)x7! xmod(1)where for every i> 1, x","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"289-301"},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000035","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411063","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}
引用次数: 8
Minimal genus and fibering of canonical surfaces via disk decomposition 通过圆盘分解的正则曲面的最小属和纤维化
Lms Journal of Computation and Mathematics Pub Date : 2014-01-01 DOI: 10.1112/S1461157013000272
A. Stoimenow
{"title":"Minimal genus and fibering of canonical surfaces via disk decomposition","authors":"A. Stoimenow","doi":"10.1112/S1461157013000272","DOIUrl":"https://doi.org/10.1112/S1461157013000272","url":null,"abstract":"This paper contains some applications of the description of knot diagrams by genus, and Gabai’s methods of disk decomposition. We show that there exists no genus one knot of canonical genus 2, and that canonical genus 2 fiber surfaces realize almost every Alexander polynomial only finitely many times (partially confirming a conjecture of Neuwirth).","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"77-108"},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157013000272","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411282","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}
引用次数: 4
Computation of Mordell–Weil bases for ordinary elliptic curves in characteristic two 特征二的普通椭圆曲线的modell - weil基的计算
Lms Journal of Computation and Mathematics Pub Date : 2014-01-01 DOI: 10.1112/S1461157014000175
G. Moehlmann
{"title":"Computation of Mordell–Weil bases for ordinary elliptic curves in characteristic two","authors":"G. Moehlmann","doi":"10.1112/S1461157014000175","DOIUrl":"https://doi.org/10.1112/S1461157014000175","url":null,"abstract":"In this paper we consider ordinary elliptic curves over global function fields of characteristic 2. We present a method for performing a descent by using powers of the Frobenius and the Verschiebung. An examination of the local images of the descent maps together with a duality theorem yields information about the global Selmer groups. Explicit models for the homogeneous spaces representing the elements of the Selmer groups are given and used to construct independent points on the elliptic curve. As an application we use descent maps to prove an upper bound for the naive height of an S-integral point on A. To illustrate our methods, a detailed example is presented.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"1-13"},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000175","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411366","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}
引用次数: 1
A note on uniform approximation of functions having a double pole 关于双极函数的一致近似的注释
Lms Journal of Computation and Mathematics Pub Date : 2014-01-01 DOI: 10.1112/S1461157013000387
I. Moale, V. Pillwein
{"title":"A note on uniform approximation of functions having a double pole","authors":"I. Moale, V. Pillwein","doi":"10.1112/S1461157013000387","DOIUrl":"https://doi.org/10.1112/S1461157013000387","url":null,"abstract":"We consider the classical problem of finding the best uniform approximation by polynomials of $1/(x-a)^2,$ where $a>1$ is given, on the interval $[-! 1,1]$ . First, using symbolic computation tools we derive the explicit expressions of the polynomials of best approximation of low degrees and then give a parametric solution of the problem in terms of elliptic functions. Symbolic computation is invoked then once more to derive a recurrence relation for the coefficients of the polynomials of best uniform approximation based on a Pell-type equation satisfied by the solutions.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"233-244"},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157013000387","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411539","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}
引用次数: 1
Subexponential class group and unit group computation in large degree number fields 大次数域的次指数类群和单位群计算
Lms Journal of Computation and Mathematics Pub Date : 2014-01-01 DOI: 10.1112/S1461157014000345
Jean-François Biasse, C. Fieker
{"title":"Subexponential class group and unit group computation in large degree number fields","authors":"Jean-François Biasse, C. Fieker","doi":"10.1112/S1461157014000345","DOIUrl":"https://doi.org/10.1112/S1461157014000345","url":null,"abstract":"We describe how to compute the ideal class group and the unit group of an order in a number field in subexponential time. Our method relies on the generalized Riemann hypothesis and other usual heuristics concerning the smoothness of ideals. It applies to arbitrary classes of number fields, including those for which the degree goes to infinity.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"385-403"},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000345","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411913","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}
引用次数: 56
Class numbers of real cyclotomic fields of composite conductor 复合导体实环切场的类数
Lms Journal of Computation and Mathematics Pub Date : 2014-01-01 DOI: 10.1112/S1461157014000382
John C. Miller
{"title":"Class numbers of real cyclotomic fields of composite conductor","authors":"John C. Miller","doi":"10.1112/S1461157014000382","DOIUrl":"https://doi.org/10.1112/S1461157014000382","url":null,"abstract":"Until recently, the ‘plus part’ of the class numbers of cyclotomic fields had only been determined for fields of root discriminant small enough to be treated by Odlyzko’s discriminant bounds.However, by finding lower bounds for sums over prime ideals of the Hilbert class field, we can now establish upper bounds for class numbers of fields of larger discriminant. This new analytic upper bound, together with algebraic arguments concerning the divisibility properties of class numbers, allows us to unconditionally determine the class numbers of many cyclotomic fields that had previously been untreatable by any known method.In this paper, we study in particular the cyclotomic fields of composite conductor.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"404-417"},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000382","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411964","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}
引用次数: 11
Minimal models for -coverings of elliptic curves 椭圆曲线的最小复盖模型
Lms Journal of Computation and Mathematics Pub Date : 2014-01-01 DOI: 10.1112/S1461157014000217
T. Fisher
{"title":"Minimal models for -coverings of elliptic curves","authors":"T. Fisher","doi":"10.1112/S1461157014000217","DOIUrl":"https://doi.org/10.1112/S1461157014000217","url":null,"abstract":"In this paper we give a new formula for adding $2$ -coverings and $3$ -coverings of elliptic curves that avoids the need for any field extensions. We show that the $6$ -coverings obtained can be represented by pairs of cubic forms. We then prove a theorem on the existence of such models with integer coefficients and the same discriminant as a minimal model for the Jacobian elliptic curve. This work has applications to finding rational points of large height on elliptic curves.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"112-127"},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000217","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411399","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}
引用次数: 0
Finding roots in Fpn with the successive resultants algorithm 用连续结果算法求Fpn的根
Lms Journal of Computation and Mathematics Pub Date : 2014-01-01 DOI: 10.1112/S1461157014000138
C. Petit
{"title":"Finding roots in Fpn with the successive resultants algorithm","authors":"C. Petit","doi":"10.1112/S1461157014000138","DOIUrl":"https://doi.org/10.1112/S1461157014000138","url":null,"abstract":"The problem of solving polynomial equations over finite fields has many applications in cryptography and coding theory. In this paper, we consider polynomial equations over a 'large' finite field with a 'small' characteristic. We introduce a new algorithm for solving this type of equations, called the successive resultants algorithm (SRA). SRA is radically different from previous algorithms for this problem, yet it is conceptually simple. A straightforward implementation using Magma was able to beat the built-in Roots function for some parameters. These preliminary results encourage a more detailed study of SRA and its applications. Moreover, we point out that an extension of SRA to the multivariate case would have an important impact on the practical security of the elliptic curve discrete logarithm problem in the small characteristic case. Supplementary materials are available with this article. © 2014 The Author.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"203-217"},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157014000138","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411254","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}
引用次数: 5
A symmetric non-stationary subdivision scheme 一种对称非平稳细分方案
Lms Journal of Computation and Mathematics Pub Date : 2014-01-01 DOI: 10.1112/S1461157013000375
S. Siddiqi, M. Younis
{"title":"A symmetric non-stationary subdivision scheme","authors":"S. Siddiqi, M. Younis","doi":"10.1112/S1461157013000375","DOIUrl":"https://doi.org/10.1112/S1461157013000375","url":null,"abstract":"This paper proposes a new family of symmetric 4-point ternary non-stationary subdivision schemes that can generate the limit curves of C 3 continuity. The continuity of this scheme is higher than the existing 4-point ternary approximating schemes. The proposed scheme has been developed using trigonometric B-spline basis functions and analyzed using the theory of asymptotic equivalence. It has the ability to reproduce or regenerate the conic sections, trigonometric polynomials and trigonometric splines as well. Some graphical and numerical examples are being considered, by choosing an appropriate tension parameter 0 < α < π/ 3, to show the usefulness of the proposed scheme. Moreover, the H¨older regularity and the reproduction property are also being calculated.","PeriodicalId":54381,"journal":{"name":"Lms Journal of Computation and Mathematics","volume":"17 1","pages":"259-272"},"PeriodicalIF":0.0,"publicationDate":"2014-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1112/S1461157013000375","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"63411530","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}
引用次数: 3
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