Project/Area Number  10640049 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Algebra

Research Institution  KINKI UNIVERSITY 
Principal Investigator 
IZUMI Shuzo Kinki Univ., Dept. Sci. and Tech., professor, 理工学部, 教授 (80025410)

CoInvestigator(Kenkyūbuntansha) 
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)

Project Fiscal Year 
1998 – 1999

Project Status 
Completed(Fiscal Year 1999)

Budget Amount *help 
¥1,900,000 (Direct Cost : ¥1,900,000)
Fiscal Year 1999 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 1998 : ¥1,100,000 (Direct Cost : ¥1,100,000)

Keywords  zeroestimate / order / Noetherian function / Fukui invariant / 零評価 / 位数 / ネタリアン関数 / 福井の不変量 / 局所環 
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 'blowanalyticity'. 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.
