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The real field with convergent generalized power series. (English) Zbl 0905.03022
An 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 $$[0,\infty)$$.

MSC:
 03C50 Models with special properties (saturated, rigid, etc.) 26E05 Real-analytic functions
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