o-minimal structures: low arity versus generation

Abstract

We show that an analogue of 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 $\mathbb{R}^{n+1}$.