Polynomial reformulation of the Kuo criteria for v-sufficiency of map-germs

Details Polynomial reformulation of the Kuo criteria for v-sufficiency of map-germs Polynomial reformulation of the Kuo criteria for v-sufficiency of map-germs

2009-07-03

arxiv.org/abs/0907.0571v3

Discrete and Continuous Dynamical Systems Series B, 14, No. 2, September 2010, 587-602

Mathematics

Algebraic Geometry

16 pages, 25 bibliography references, corrected misprints Accepted for publication in DCDS-B, added acknowledgments to the sup

Scientific paper

10.3934/dcdsb.2010.14.587

In the paper a set of necessary and sufficient conditions for \textit{v-}sufficiency (equiv. \textit{sv-}sufficiency) of jets of map-germs $f:(\mathbb{R}^{n},0)\to (\mathbb{R}^{m},0)$ is proved which generalize both the Kuiper-Kuo and the Thom conditions in the function case ($m=1$) so as the Kuo conditions in the general map case ($m>1$). Contrary to the Kuo conditions the conditions proved in the paper do not require to verify any inequalities in a so-called horn-neighborhood of the (a'priori unknown) set $f^{-1}(0)$. Instead, the proposed conditions reduce the problem on \textit{v-}sufficiency of jets to evaluating the local {\L}ojasiewicz exponents for some constructively built polynomial functions.

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