Characterisations of V-sufficiency and $C^0$-sufficiency of relative jets
Résumé
The Kuiper-Kuo theorem is well-known in the singularity theory, as a result giving sufficient conditions for r-jets to be V-sufficient and $C^0$-sufficient in $C^r$ functions or $C^{r+1}$ functions. The converse is also proved by J. Bochnak - S. Lojasiewicz. Generalisations of the criteria for V-sufficiency and $C^0$-sufficiency to the mapping case are established by T.-C. Kuo and J. Bochnak - W. Kucharz, respectively. In this paper we consider the problems of sufficiency of jets relative to a given closed set. We give several equivalent conditions to the relative Kuo condition, and discuss characterisations of V-sufficiency and $C^0$-sufficiency of relative jets, 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.