Open Access
October, 2007 Non-smooth points set of fibres of a semialgebraic mapping
Satoshi KOIKE, Masahiro SHIOTA
J. Math. Soc. Japan 59(4): 953-969 (October, 2007). DOI: 10.2969/jmsj/05940953


For a semialgebraic mapping between semialgebraic sets, we consider the set of points at which the fibre is not smooth. In this paper we discuss whether the singular set is itself semialgebraic, when it has codimension bigger than or equal to 2 in the domain of f and whether the mapping is semialgebraically trivial along the smooth part of the fibre, giving several examples which show optimality of those results. In addition, we give an example of a polynomial function f such that even the ( a f ) condition in the weak sense fails in a neighbourhood of a smooth fibre, but f is semialgebraically trivial along it.


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Satoshi KOIKE. Masahiro SHIOTA. "Non-smooth points set of fibres of a semialgebraic mapping." J. Math. Soc. Japan 59 (4) 953 - 969, October, 2007.


Published: October, 2007
First available in Project Euclid: 10 December 2007

zbMATH: 1166.57017
MathSciNet: MR2369999
Digital Object Identifier: 10.2969/jmsj/05940953

Primary: 57R45
Secondary: 14P10 , 14P20

Keywords: Nash mapping , semialgebraic geometry , Thom condition $(a_{f})$

Rights: Copyright © 2007 Mathematical Society of Japan

Vol.59 • No. 4 • October, 2007
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