Differentiation in P-minimal structures and a p-adic Local Monotonicity Theorem

TL;DR

This paper establishes a p-adic local Monotonicity Theorem for P-minimal structures, demonstrating that definable functions are almost everywhere strictly differentiable and satisfy the Local Jacobian Property, advancing understanding of p-adic geometry.

Contribution

It proves a p-adic local Monotonicity Theorem for P-minimal structures, confirming a conjecture and analyzing derivatives within these structures.

Findings

Definable functions are almost everywhere strictly differentiable.

Functions satisfy the Local Jacobian Property.

The theorem applies to a broad class of P-minimal structures.

Abstract

We prove a p-adic, local version of the Monotonicity Theorem for P-minimal structures. The existence of such a theorem was originally conjectured by Haskell and Macpherson. We approach the problem by considering the first order strict derivative. In particular, we show that, for a wide class of P-minimal structures, the definable functions f : K -> K are almost everywhere strictly differentiable and satisfy the Local Jacobian Property.