The real field with convergent generalized power series.

*(English)*Zbl 0905.03022An expansion of the ordered real field \(\mathbb R\) is called o-minimal if every set of reals definable in the expansion is a finite union of points and intervals. Following their introduction in the early 80s by L. van den Dries and, in a more general setting, by A. Pillay and C. Steinhorn, o-minimality was found to provide an elegant and surprisingly efficient generalization of semialgebraic and subanalytic geometry [see L. van den Dries, Tame topology and o-minimal structures (Lond. Math. Soc. Lect. Note Ser. 248) (1998)]. Examples of proper o-minimal expansions of the reals are (i) \(\mathbb R_{\text{an}}\), the expansion of \(\mathbb R\) by restricted analytic functions (J. Denef and L. van den Dries, 1986), (ii) \(\mathbb R_{\text{exp}}\), the expansion of \(\mathbb R\) by exponentiation (A. Wilkie, 1991), and, moreover, (iii) \(\mathbb R_{\text{an,exp}}\), the expansion of \(\mathbb R\) by both restricted analytic functions and exponentiation (L. van den Dries, A. Macintyre, D. Marker, 1994). Recently, building o-minimal expansions of the real field has become a very lively activity. In the paper under review the authors develop a new way to prove model completeness and o-minimality of expansions of \(\mathbb R\), and apply this to a particular expansion \(\mathbb R_{\text{an}^*}\), for which the previous methods fail. Inductive arguments using blow-up maps as in works of J.-Cl. Tougeron are an important ingredient of the proof; also, ideas of A. Gabrielov are crucial. Every relation that is definable in \(\mathbb R_{\text{an}}\) is definable in \(\mathbb R_{\text{an}^*}\) as well. However, there are functions that are definable in \(\mathbb R_{\text{an}^*}\) but not in \(\mathbb R_{\text{an}}\). For example, the function \(x\mapsto\sum_{n=1}^\infty x^{\log n}\) with domain \([0,e^{-2}]\) is definable in \(\mathbb R_{\text{an}^*}\), but not definable even in \(\mathbb R_{\text{an,exp}}\). The expansion \(\mathbb R_{\text{an}^*}\) is shown to be polynomially bounded; the latter means that every function definable in the expansion is majorized by a polynomial. The structure \(\mathbb R_{\text{an}^*}\) is defined to be the expansion of \(\mathbb R\) by all functions \(f:\mathbb R^m\to\mathbb R\) (for all natural numbers \(m\)) that are 0 outside \([0,1]^m\) and are given on \([0,1]^m\) by a formal power series \(\sum_{\alpha}c_{\alpha}X^\alpha\) with \(\sum_{\alpha}| c_{\alpha}| r^\alpha<\infty\) for some polyradius \(r=(r_1,\dots ,r_m)\), where \(r_1,\dots ,r_m>1\). Here the multi-index \(\alpha=(\alpha_1,\dots ,\alpha_m)\) ranges over \([0,\infty)^m\), the coefficients \(c_\alpha\) are real, \(X^\alpha\) denotes the formal monomial \(X_1^{\alpha_1}\dots X_m^{\alpha_m}\), and the set \(\{\alpha : c_\alpha\neq 0\}\), the support of the series, is contained in \(S_1\times\dots\times S_m\) for some well ordered subsets \(S_i\) of \([0,\infty)\).

Reviewer: O.V.Belegradek (Kemerovo)