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Number Theory and its Development to Discrete Mathematics

数论及其向离散数学的发展

基本信息

批准号：
20540019
负责人：
NAKAHARA Toru
金额：
$ 2.83万
依托单位：
Saga University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2008
资助国家：
日本
起止时间：
2008 至 2010
项目状态：
已结题

项目摘要

On the core subjects of this research theme ; Number Theory, specifically Hasse's problem related to Abelian fields[A], Arithmetic, Algebraic geometry[B] and its application to Discrete Mathematics[C], we held the Workshop on Number Theory in Saga in each August and January of 2008~2010. The research organizer stayed at NUCES during two and half years to work the joint research with PhD scholars at both campuses. On Hasse's problem, the organizer obtained the characterization on the monogeneity for certain family of pure sextic and pure octic fields by the joint work with PhD scholars at NUCES. In our research on algebraic geometry codes, Cooperative Uehara gave an idea of constructing a new class of algebraic geometry codes different from the known codes of one-point type. Also, applying the concept of evaluation codes, which generalize one-point-type algebraic geometry codes, Uehara invented a method of constructing codes from integer rings on algebraic number fields, and presented s … More ome explicit examples of such codes[C]. Cooperative Miyazaki studied the minimal free resolution of Buchsbaum varieties and obtained a classification of the Buchsbaum variety in terms of the Castelnuovo-Mumford regularity[B]. Cooperative Terai studied Stanly-Reisner ideals, which are squarefree monomial ideals in polynomial rings[B]. Cooperative Katayama . has determined finite symplectic groups of cube and 4th order, using the structure of the unit groups of cubic and quadratic fields, and announced these results at the workshop in Saga 2011. Newman, Shanks and Williams determined finitesymplectic groups of square order in1980's. Katayama also investigated the number of congruent k-polygons inscribed in a unit circle, where the vertices chosen from n division points of the circle[C]. Cooperative Taguchi studied the ramification theory of truncated discrete valuation rings (= : tdvr's) and the (non)existence of mod p Galois representations. On the former, Taguchi proved (jointly with T. Hiranouchi) that the category of finite extensions of a tdvr A is equivalent to a category of finite extensions, with restricted ramification, of a complete discrete valuation field which lifts A. On the latter, Taguchi proved (jointly with H. Moon) the non-existence of 2-dimensional mod 2 Galois representations for some quadratic fields[B]. Less

关于本研究主题的核心课题：数论，特别是哈塞问题与阿贝尔领域[A]，算术，代数几何[B]及其在离散数学中的应用[C]，我们在2008年8月和2010年1月在佐贺举办了数论研讨会。研究组织者在NUCES呆了两年半，与两个校区的博士学者一起进行联合研究。在Hasse问题上，组织者通过与NUCES博士学者的合作，获得了某些纯六次和纯八次场族的单整性刻画。在我们对代数几何码的研究中，Cooperative Uehara提出了构造一类不同于已知单点型码的新的代数几何码的思想。此外，上原还应用了赋值码的概念，推广了单点型代数几何码，发明了一种由代数数域上的整数环构造码的方法，并给出了一种由代数数域上的整数环构造码的方法。关于我们这类码的一些显式例子[C]. Cooperative宫崎研究了Buchsbaum簇的最小自由归结，并根据Castelnuovo-Mumford正则性得到了Buchsbaum簇的分类[B]。合作寺井研究了Stanly-Reisner理想，这是多项式环中的平方自由单项式理想[B]。合作片山。利用三次和二次域的单位群的结构，确定了立方和四阶的有限辛群，并在佐贺2011年的研讨会上宣布了这些结果。纽曼，Shanks和威廉姆斯在20世纪80年代确定了平方阶有限辛群。片山还研究了内接于单位圆的全等k-多边形的数量，其中顶点选自圆的n个分割点[C]。合作田口研究了截断离散赋值环（=：tdvr的）的分支理论和mod p伽罗瓦表示的（不）存在性。关于前者，田口证明（与T。Hiranouchi）证明了tdvr A的有限扩张范畴等价于提升A的完备离散赋值域的有限扩张范畴，其具有限制分支.关于后者，田口证明（与H。Moon）某些二次域的2维模2伽罗瓦表示的不存在性[B].少