2002 Fiscal Year Final Research Report Summary
STUDY OF SEMIGROUPS OF SETS OF VALUATIONS DEFINED BY EXCEPTIONAL COMPONENTS
| Project/Area Number |
12640050
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | KINKI UNIVERSITY |
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Principal Investigator |
IZUMI Shuzo KINKI UNIVERSITY, SCHOOL of SCIENCE and ENGINEERING, PROFESSOR, 理工学部, 教授 (80025410)
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| Co-Investigator(Kenkyū-buntansha) |
KOIKE Satoshi DEPT. of MATHEMATICS, HYOGO UNIVERSITY of TEACHER EDUCATION, ASSOCIATE PROFESSOR, 教育学部, 助教授 (60161832)
FUKUI Toshizumi SAITAMA UNIVERSITY, FACULTY of SCIENCE, PROFESSOR, 理学部, 教授 (90218892)
NAGAOKA Shoyu KINKI UNIVERSITY, SCHOOL of SCIENCE and ENGINEER, PROFESSOR, 理工学部, 教授 (20164402)
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| Project Period (FY) |
2000 – 2002
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| Keywords | order of functions / Spallek function / Taylor expansion / Fukui invariant / flatness of a differentiable functions |
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| Research Abstract |
Seeing Kei-iti Watanabe's announcement of a result closely related to the our project, we were obliged to shift our theme a little.
Thus the first result is finding method of computation and proving stability of Fukui invariant. Fukui invariant is important to test blow-analytic equivalence. It is defined for a singularity as the set of values of valuations defined by curves running on it. We obtained our results reducing the problem to that of valuations defined by exceptional components.
Next result is observation of the relation of order and values on a closed set. Let X be a closed subset of Euclidean space adherent to the origin. If X is fat in a certain sense and if differentiable function f rapidly decrease on X, then Maclaurin expansion of f vanishes upto high order terms. In typical case, the order of expansion and the order of decreasing is linearly comparable. We found a converging point sequence X whose orders are linearly comparable. These results are closely related to the old theorem of the head researcher on the valuation of analytic local algebra. We also found an example of X such that f|X is inefficient to control expansion using Osgood's map germ.
We also proved a zero estimate theorem for Noetherian functions as a corollary of Gabrielov's theorem. This treats Noetherian functions and hence generalizes classical zero estimate theorems obtained by number theorists.
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