O-minimal structures: low arity versus generation

TL;DR

The paper demonstrates that in the real subanalytic setting, the o-minimal structure generated by restricted analytic functions in n variables is strictly smaller than the entire structure of global subanalytic sets, contrasting with their equivalence in n+1 variables.

Contribution

It establishes a fundamental difference in the generative power of restricted analytic functions in low arity versus the full subanalytic structure, showing an analogue of Hilbert's Thirteenth Problem fails.

Findings

Generated o-minimal structures are strictly smaller in n variables

Both structures coincide in n+1 variables

Highlights limitations of restricted analytic functions in low arity

Abstract

We show that an analogue of the Hilbert's Thirteenth Problem fails in the real subanalytic setting.Namely we prove that, for any integer $n$, the o-minimal structure generated by restricted analytic functions in $n$ variables is strictly smaller than the structure of all global subanalytic sets, whereas these two structures define the same subsets in $\mathbb R ^{n+1}$.