Approximations and Lipschitz continuity in p-adic semi-algebraic and subanalytic geometry

TL;DR

This paper proves that p-adic semi-algebraic functions with local Lipschitz constant 1 are piecewise Lipschitz continuous with the same constant, using fine approximation techniques involving monomials with fractional exponents.

Contribution

It introduces new approximation results for p-adic semi-algebraic functions, enabling the transfer of local Lipschitz continuity to a piecewise setting with the same Lipschitz constant.

Findings

Functions can be approximated by monomials with fractional exponents

Derivative of the monomial approximates the derivative of the function

Results extend to parametrized and subanalytic cases

Abstract

It was already known that a p-adic, locally Lipschitz continuous semi-algebraic function is piecewise Lipschitz continuous, where the pieces can be taken semi-algebraic. We prove that if the function has locally Lipschitz constant 1, then it is also piecewise Lipschitz continuous with the same Lipschitz constant 1. We do this by proving the following fine preparation results for p-adic semi-algebraic functions in one variable. Any such function can be well approximated by a monomial with fractional exponent such that moreover the derivative of the monomial is an approximation of the derivative of the function. We also prove these results in parametrized versions and in the subanalytic setting.