Differentiability Properties of Log-Analytic Functions

Abstract

We show that the derivative of a log-analytic function is log-analytic. We prove that log-analytic functions exhibit strong quasianalytic properties. We establish the parametric version of Tamm's theorem for log-analytic functions.

On Preparation Theorems for $\mathbb{R}_{an,exp}$-definable functions

“…Theorem A is strong enough to obtain nice differentiability properties for loganalytic functions like non-flatness or a parametric version of Tamm's theorem (compare with [6]): With property (2) above one sees that theorem A coincides with Theorem 2.30 in [6], a preparation theorem for log-analytic functions on simple cells.…”

“…(2) Suppose that for every t ∈ π(C) there is d t ∈ R >0 ∪ {∞} such that C t = ]0, d t [ (such a cell C is called simple, compare with [6]). In [6] it is shown that Θ j = 0.…”

“…Suppose the contrary. Let E be a set of C-heirs such that h can be constructed from E. An easy calculation shows that the center of every logarithmic scale on C vanishes (compare with Definition 2.1(d) or Proposition 2.19 in [6]). So we see E = ∅ or E = 1.…”

“…Call a definable cell C ⊂ R n × R simple if C t is of the form ]0, d t [ for every t ∈ π(C) where d t ∈ R >0 . Proposition 2.19 in [6] states that the center of every logarithmic scale on such a simple cell C vanishes. This gives the following.…”

“…We refer to the book of Van den Dries [3] for the general properties of o-minimal structures; in the preliminary section we state the definition and give examples. Log-analytic functions have been defined by Lion and Rolin in their seminal paper [9] (see also [6] for the definition). They are iterated compositions from either side of globally subanalytic functions (see [2]) and the global logarithm.…”