O-minimal structures: low arity versus generation

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 $\mathbb R ^{n+1}$.