Examples in the theory of sufficiency of jets

Kucharz, Wojciech

doi:10.1090/S0002-9939-1986-0813830-1

Examples in the theory of sufficiency of jets
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Proc. Amer. Math. Soc. 96 (1986), 163-166 Request permission

Abstract:

It is shown that for a given nonnegative integer $s$, there exist a positive integer $r(s)$ and an $r(s)$-jet ${\upsilon _s}$ with source at $O \in {{\mathbf {R}}^3}$ which is not $V$-sufficient in the class of ${C^{r(s) + s}}$-realizations and is ${C^0}$-sufficient in the class of ${C^{r(s) + s + 1}}$-realizations. In the complex case, a jet with source at $O \in {{\mathbf {C}}^2}$ which is $V$-sufficient but not ${C^0}$-sufficient in the class of holomorphic realizations is constructed.

References

Normal forms in neighborhoods of degenerate critical points

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