Growth of Log-Analytic Functions

TL;DR

This paper investigates the growth behavior of log-analytic functions, establishing polynomial bounds in one dimension and exponential growth in higher dimensions, with specific bounds on definable sets.

Contribution

It provides new bounds on the growth of log-analytic functions, showing polynomial bounds in one variable and exponential growth in multiple variables, with conditions on definable sets.

Findings

Unary log-analytic functions are polynomially bounded.

Higher-dimensional log-analytic functions can exhibit exponential growth.

Log-analytic functions are polynomially bounded on certain definable sets.

Abstract

We show that unary log-analytic functions are polynomially bounded. In the higher dimensional case globally a log-analytic function can have exponential growth. We show that a log-analytic function is polynomially bounded on a definable set which contains the germ of every ray at infinity.