1999 Fiscal Year Final Research Report Summary
Research of rings defined by a system of differential equations
| Project/Area Number |
10640049
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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 Univ., Dept. Sci. and Tech., professor, 理工学部, 教授 (80025410)
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| Co-Investigator(Kenkyū-buntansha) |
FUKUI Toshizumi Saitama Univ., Dept. Sci., assistant professor, 理学部, 助教授 (90218892)
AOKI Takashi Kinki Univ., Dept. Sci. and Tech., professor, 理工学部, 教授 (80159285)
NAGAOKA Shoyu Kinki Univ., Dept. Sci. and Tech., professor, 理工学部, 教授 (20164402)
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| Project Period (FY) |
1998 – 1999
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| Keywords | zero-estimate / order / Noetherian function / Fukui invariant |
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| Research Abstract |
We are interested in vanishing order of functions at singularity. First we investigated Noetherian functions, which were found and studied by Khovanskii, Tougeron and Gabrielov. But Gabrielov and Khovanskii were sooner to publish a result concerning some estimate of multiplicity which is closely related to our purpose.
Hence we shifted our purpose to the study of Fukui invariant. It is the set of the orders of restrictions to curves of a fixed function germ. This invariant is important for the study of 'blow-analyticity'. Using resolution of zero locus of the function, we reduced the problem to elementary number theory. Together with S. Koike and T.-C. Kuo, we found a convenient complete method to compute Fukui invariant and settled problems on its stability and periodicity. This manuscript is now submitted to a journal.
I have given 2 talks at RIMS, Kyoto University on general theory on orders of functions at singularity.
Meanwhile, theory of order is analogous to that of convergence. More than ten years ago T. Ohsawa posed me a problem on convergence of formal morphisms of completions of complex spaces along Moishezon subspaces. Recently I have completed a paper on his problem. The assertion is that such a morphism converges either everywhere or nowhere under a certain reasonable condition.
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