On the real exponential field with restricted analytic functions (Q1320026)

The present paper generalizes a result of A. Wilkie who has proved the model-completeness of the structure \(\mathbb{R}_{\exp}\), i.e. the field of reals together with exponentiation. In an earlier paper, Denef and van den Dries have already proved the model-completeness of \(\mathbb{R}_{\text{an}}\), the field of reals endowed for every \(m\geq 1\) with the restriction to \([- 1,+ 1]^ m\) of all \(m\)-ary real-valued functions \(f\), analytic in a neighbourhood of \([- 1, +1]^ m\). Now, this paper gives a proof of the model-completeness of the structure \((\mathbb{R}_{\text{an}},\exp)\), i.e. \(\mathbb{R}_{\text{an}}\) together with real exponentiation. The proof generalizes that of A. Wilkie and, in addition, gives o-minimality for the structure \((\mathbb{R}_{\text{an}}, \exp)\). There is a more recent work of the first author, \textit{A.Macintyre} and \textit{D. Marker} [Ann. Math., II. Ser. 140, 183-205 (1994)] which proves an even stronger result for \((\mathbb{R}_{\text{an}}, \exp)\) by using a different method in the proof.