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

## TL;DR

This paper extends the theory of jet sufficiency to the relative case, providing characterisations and examples that distinguish between V-sufficiency and C^0-sufficiency for polynomial functions.

## Contribution

It develops new characterisations of V-sufficiency and C^0-sufficiency for relative jets, and constructs examples illustrating differences between these notions.

## Key findings

- V-sufficiency and C^0-sufficiency are equivalent in the non-relative case.
- The paper provides characterisations of relative finite V-determinacy.
- Examples show relative r-jets can be V-sufficient but not C^0-sufficient.

## Abstract

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.