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September  2010, 14(2): 587-602. doi: 10.3934/dcdsb.2010.14.587

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

Victor Kozyakin 1,

1. 

Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoj Karetny lane 19, Moscow 127994 GSP-4

Received  July 2009 Published  June 2010

In the paper a set of necessary and sufficient conditions for v-sufficiency (equiv. 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 v-sufficiency of jets to evaluating the local Łojasiewicz exponents for some constructively built polynomial functions.
Keywords: Łojasiewicz exponent, sufficiency of map-germs, critical points., Finite determinedness, truncated equations.
Mathematics Subject Classification: Primary: 58K40; 58K45; Secondary: 32S1.
Citation: Victor Kozyakin. Polynomial reformulation of the Kuo criteria for v- sufficiency of map-germs. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 587-602. doi: 10.3934/dcdsb.2010.14.587
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