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

TL;DR

This paper introduces polynomial-based conditions for v-sufficiency of map-germs that generalize existing criteria and avoid complex neighborhood inequalities, simplifying the verification process.

Contribution

It provides new necessary and sufficient polynomial reformulations of v-sufficiency conditions that extend previous Kuiper-Kuo and Thom criteria.

Findings

Conditions do not require inequalities in horn-neighborhoods.

Reduces v-sufficiency verification to evaluating polynomial Lojasiewicz exponents.

Generalizes criteria to higher-dimensional map-germs.

Abstract

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.