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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Determinacy of smooth germs with real isolated line singularities


Authors: Bohao Sun and Leslie C. Wilson
Journal: Proc. Amer. Math. Soc. 129 (2001), 2789-2797
MSC (1991): Primary 58K40; Secondary 32S05
DOI: https://doi.org/10.1090/S0002-9939-01-06068-3
Published electronically: February 9, 2001
MathSciNet review: 1838804
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Abstract:

The germ of a smooth real-valued function on Euclidean space is called a real isolated line singularity if its singular set is a nonsingular curve, its Jacobian ideal is \Lojasiewicz at the singular set, and its Hessian determinant restricted to the singular set is \Lojasiewicz at 0. Consider the set of all germs whose singular set contains a fixed nonsingular curve $L$. We prove that such a germ $f$ is infinitely determined among all such germs with respect to composition by diffeomorphisms preserving $L$ if, and only if, the Jacobian ideal of $f$ contains all germs which vanish on $L$ and are infinitely flat at 0 if, and only if, $f$ is a real isolated line singularity.


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Additional Information

Bohao Sun
Affiliation: Department of Mathematics, University of Hawaii, Honolulu, Hawaii 96822

Leslie C. Wilson
Affiliation: Department of Mathematics, University of Hawaii, Honolulu, Hawaii 96822
Email: les@math.hawaii.edu

DOI: https://doi.org/10.1090/S0002-9939-01-06068-3
Keywords: Non-isolated singularities, line singularities, infinite determinacy, Lojasiewicz inequality
Received by editor(s): January 5, 2000
Published electronically: February 9, 2001
Communicated by: Jozef Dodziuk
Article copyright: © Copyright 2001 American Mathematical Society