2010 Fiscal Year Final Research Report
Number Theory and its Development to Discrete Mathematics
| Project/Area Number |
20540019
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
Algebra
|
|---|
| Research Institution | Saga University |
|---|
Principal Investigator |
NAKAHARA Toru Saga University, 佐賀大学, 名誉教授 (50039278)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
UEHARA Tsuyoshi 佐賀大学, 大学院・工学系研究科, 教授 (80093970)
MIYAZAKI Chikashi 佐賀大学, 大学院・工学系研究科, 教授 (90229831)
TERAI Naoki 佐賀大学, 文化教育学部, 准教授 (90259862)
KATAYAMA ShinーIchi 徳島大学, 総合科学部, 教授 (70194777)
TAGUCHI Yuuichiro 九州大学, 大学院・数理学研究院, 准教授 (90231399)
|
|---|
| Research Collaborator |
CLAUDE Levesque ラバル大学, 教授
KIM HyunKuang POSTEC, 教授
SYED Inayat Ali Shah Islamia College University, 教授
|
|---|
| Project Period (FY) |
2008 – 2010
|
| Keywords | 代数的整数論 / Hasseの問題 / 純6次体 / 頂切離散付値環 / Buchsbaum多様体 / モノミアル イデアル / 一点型代数幾何符号 / symplectic group / 分割関数 |
|---|
| Research Abstract |
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
|
