Experimental Method of Mathematics

数学实验方法

• 批准号: 08404008
• 负责人: YAMAMOTO Yoshihiko
• 金额: $ 12.61万
• 依托单位: Osaka University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A)
• 财政年份: 1996
• 资助国家: 日本
• 起止时间: 1996 至 1997
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08404008/
• 关键词: Computer algbra; Limit-formula; Elliptic curve; Taniyama-Shumura conjecture; Automorphic function; Class fields; Riemann surface; 実験数学; ゼータ関数; 複素関数の表示法; 二次体の類数; 虚数乗法; ヤコビ多様体; 加法公式; 等分方程式

We worked, by usisng softwares for the computer algebra, manily in the fields of arithmetics, algebraic geometry, automorphic functions, Riemann surfaces and cryptosystems and gots many interesting results, some of them are as follows :1. Algebraic Number Thoery : (1) Computation of classnumbers and fundamental units of number fields. (2) Relations between certain residule class groups of real quadratic fields and class groups of imaginary quadratic fields. (3) Numerical computations of Kronecker's limit-formula of real quadratic fields. (4) Group structures of the unit groups of Dihedral extensions over the rational number field.2. Arithmetics in elliptic curves defined over the rational field : (1) Existence of the canonical parameters and canonical power series solutions, (2) Uniformizaton of elliptic curves by automorphic functions. (3) Algorithmic Computation of canonical power series and the global zeta-function over the ring of rational integers. (4) Existence of canonical arithmetic-structures in elliptic curves.3. Application of 3D-graphical representation of complex functions : (1) Represatations of Riemann surfaces by Complex Plot commands. (2) Investigating periodical properties of complex functions given by power series. (3) Automorphic properties of canonical power seires of elliptic curves.

我们利用计算机代数软件，主要在算术、代数几何、自同构函数、黎曼曲面和密码系统等领域进行了工作，得到了许多有趣的结果，其中一些结果如下： 1．代数数论：（1）类数和数域基本单位的计算。 (2)实二次域的某些留数类群与虚二次域的类群之间的关系。 (3)实二次域克罗内克极限公式的数值计算。 (4)有理数域上二面体扩张的单位群的群结构。2．在有理域上定义的椭圆曲线中的算术：（1）规范参数和规范幂级数解的存在性，（2）自守函数对椭圆曲线的统一。 (3) 有理整数环上的正则幂级数和全局 zeta 函数的算法计算。 (4)椭圆曲线中规范算术结构的存在性。3．复函数 3D 图形表示的应用： (1) 通过 Complex Plot 命令表示黎曼曲面。 (2) 研究幂级数给出的复函数的周期性质。 (3)椭圆曲线正则幂系列的自守性质。

项目成果

Y.Komatsu, et al.: "A period-doubling biforcation for the Dutting equation" Osaka J.of Math.34. 605-627 (1997)

Y.Komatsu 等人：“Dutting 方程的倍周期分叉”Osaka J.of Math.34。

E.D.Negri et al.: "Gorenstein Algebra of Veronese Type" Journal of Algebra. 193. 629-639 (1997)

E.D.Negri 等人：“Gorenstein Algebra of Veronese Type”代数杂志。

K.Hirachi and Gen Komatsu: "Invariant theory of the Beryman kernel" Advanced Studies in Pure Mathematics 25 (CR-Geometry and Overdetermined Systems). 167-220 (1997)

K.Hirachi 和 Gen Komatsu：“Beryman 核的不变理论”纯数学高级研究 25（CR 几何和超定系统）。

T.Hibi: "Gotzmann theorems for exterior algebras and combinatorics" J.Algebra. 191. 174-211 (1997)

T.Hibi：“外代数和组合学的戈兹曼定理”J.Algebra。

A.Matsumura: "A survey on stability of viscous shock protiles without genuine non-stability" AMS/IP studies in Advanced Math. 3. 319-332 (1997)

A.Matsumura：“对没有真正非稳定性的粘性冲击曲线稳定性的调查”高等数学中的 AMS/IP 研究。