Eva Leenknegt, Purdue University
“Properties of functions in P-minimal structures”
The concept of P-minimality was developed by Haskell and MacPherson as a version of o-minimality for the p-adics. This should be thought of as a ’tame’ context, where one may hope that definable functions will have better properties than general p-adic functions. For example, we will consider the problem of p-adic differentiation, and show that major problems occur if one does not restrict the class of functions under consideration. The fact that p-adic differentiation is well-behaved for P-minimal structures, can also be used to show that a local version of the monotonicity theorem holds in the P-minimal context. Another problem is the existence of Skolem functions. A P-minimal structure only has cell decomposition if it also has definable Skolem functions. However, at this moment the existence of such functions is still an open problem. I will use some observations in weaker structures to explain why such functions probably do not exist for all P-minimal structures.