Characterisations of V-sufficiency and C^0-sufficiency of relative jets

Abstract

The Kuiper-Kuo theorem ([14, 15]) is well-known in Singularity Theory, as a result giving sufficient conditions for r-jets to be V -sufficient and C 0sufficient in C r functions or C r+1 functions. The converse is also proved by J. Bochnak and S. Lojasiewicz [3]. Generalisations of the criteria for V -sufficiency and C 0 -sufficiency to the mapping case are established by T.-C. Kuo [17] and J. Bochnak and W. Kucharz [4], respectively. In this paper we consider the problems of sufficiency of jets relative to a given closed set. We give several equivalent conditions; they generalise the Kuo condition to the relative case. We also discuss characterisations of V -sufficiency and C 0 -sufficiency in the relative case, corresponding to the above non-relative results.Applying the results obtained in the relative case, we construct examples of polynomial functions whose relative r-jets are V -sufficient in C r functions and C r+1 functions but not C 0 -sufficient in C r functions and C r+1 functions, respectively. In addition, we give characterisations of relative finite V -determinacy and also relative finite C r contact determinacy.

$Γ-Σ-C^{0}-$determinacy of $Γ-Σ-C^{0}-$equivariant bifurcation problems with respect to $Γ-Σ-C^{0}-$BD and contact equivalence from the weighted point view

“…In this paper, we apply the idea of r − Σ − C 0 −equivalence map jets relative to a given closed set Σ in (R n , 0), which is considered by Karim Bekka and Satoshi Koike [1] and B. Osińska-Ulrych, T. Rodak, G. Skalski in [13], to bifurcation theory, and introduce the Γ − Σ − C 0 −bifurcation diagram and Γ − Σ − C 0 −contact equivalence about Γ−equivariant bifurcation problems from the weighted point view and explore the finite determinacy of Γ−equivariant bifurcation problems with respect to above equivalences. Some criteria about determination of Γ−equivariant bifurcation problems with respect to the Γ − Σ − C 0 −bifurcation diagram and Γ − Σ − C 0 −contact equivalence are given.…”