Two remarks on polynomially bounded reducts of the restricted analytic field with exponentiation

TL;DR

This paper explores the structure of polynomially bounded reducts of the restricted analytic field with exponentiation, providing new examples that challenge existing assumptions about definability and o-minimality.

Contribution

It introduces two novel constructions: one of o-minimal expansions with identical germs at infinity but different global functions, and another of infinitely many distinct polynomially bounded reducts.

Findings

Existence of o-minimal expansions with same germs but different global functions

Construction of infinitely many polynomially bounded reducts

Insights into the structure of restricted analytic fields with exponentiation

Abstract

This article presents two constructions motivated by a conjecture of L. van den Dries and C. Miller concerning the restricted analytic field with exponentiation. The first construction provides an example of two o-minimal expansions of a real closed field that possess the same field of germs at infinity of one-variable functions and yet define different global one-variable functions. The second construction gives an example of a family of infinitely many distinct polynomially bounded reducts (all this in the sense of definability) of the restricted analytic field with exponentiation.