Pub Date : 2014-01-01DOI: 10.1112/S1461157014000163
Ömer Küçüksakalli
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.
有一种求本源导体实环切场类数约数的学派算法。本文利用weerstrass形式给出的椭圆单位子群，对该算法的椭圆模拟进行了改进。这些椭圆单位可以用椭圆曲线上点的$def xmlpi #1{}def mathsfbi #1{boldsymbol {mathsf {#1}}}let le =leqslant let leq =leqslant let ge =geqslant let geq =geqslant def Pr {mathit {Pr}}def Fr {mathit {Fr}}def Rey {mathit {Re}}x$ -坐标表示，使我们能够在有限域上使用椭圆曲线的快速算法。
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Pub Date : 2014-01-01DOI: 10.1112/S1461157014000035
Hachem Hichri
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 Renyi [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
Boyd证明了4次Salem数的β展开式总是最终周期的。基于一个启发式论证，Boyd推测6次的Salem数也是如此，但8次的Salem数却不是。本文考察了8度的塞勒姆数，并收集了支持博伊德猜想的实验证据。1. 介绍和基本定义数字在非整数基数>1中的表示是由Renyi[11]首创的，他引入了beta展开式(也称为贪婪展开式)，用数字序列x 1 x 2 x 3:::来表示基数[0;1]区间内的任何实数，可以通过以下算法计算。贪婪算法。用c和fyg分别表示面积y的整数部分和小数部分。设r0 = xand，对于i bbbb1, xi = bbb1c, ri = fbb1g。或者，类似地，使用单位区间的变换T= T它是映射:T: [0;1] !(0, 1) x7 !Xmod(1)其中对于每一个i> 1, x
{"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}
Pub Date : 2014-01-01DOI: 10.1112/S1461157013000272
A. Stoimenow
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).
{"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}
Pub Date : 2014-01-01DOI: 10.1112/S1461157014000175
G. Moehlmann
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.
{"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}
Pub Date : 2014-01-01DOI: 10.1112/S1461157013000387
I. Moale, V. Pillwein
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.
{"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}
Pub Date : 2014-01-01DOI: 10.1112/S1461157014000345
Jean-François Biasse, C. Fieker
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.
{"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}
Pub Date : 2014-01-01DOI: 10.1112/S1461157014000382
John C. Miller
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.
{"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}
Pub Date : 2014-01-01DOI: 10.1112/S1461157014000217
T. Fisher
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.
{"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}
Pub Date : 2014-01-01DOI: 10.1112/S1461157013000375
S. Siddiqi, M. Younis
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.
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