Characterisations of \(V\)-sufficiency and \(C^0\)-sufficiency of relative jets. (English) Zbl 07810897
Summary: We consider the problems of sufficiency of jets relative to a given closed set. In the non-relative case, criteria for \(r\)-jets to be \(V\)-sufficient and \(C^0\)-sufficient in \(C^r\) mappings or \(C^{r+1}\) mappings have been obtained. In particular, it is shown that \(V\)-sufficiency and \(C^0\)-sufficiency in \(C^r\) functions or \(C^{r+1}\) functions are equivalent. In this paper we 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.
MSC:
| 57R45 | Singularities of differentiable mappings in differential topology |
| 58K40 | Classification; finite determinacy of map germs |
Keywords:
\(V\)-sufficiency of jets; \(C^0\)-sufficiency of jets; relative \(SV\)-determinacy; relative \(\mathcal{K}\) equivalenceReferences:
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