On a family of mild functions

Abstract

We prove that the function [Formula: see text] with [Formula: see text] is [Formula: see text]-mild. We apply this result to obtain a uniform [Formula: see text]-mild parametrization of the family of curves [Formula: see text] for [Formula: see text], which does not have a uniform [Formula: see text]-mild parametrization by work of Yomdin. More generally, we can parametrize families of power-subanalytic curves. This improves a result of Binyamini and Novikov that gives a [Formula: see text]-mild parametrization.