The elementary theory of restricted analytic fields with exponentiation.

*(English)*Zbl 0837.12006Let \(\mathbb{R}\{ X_1, \dots, X_m\}\) be the ring of all real power series in \(X_1, \dots, X_m\), which converge in a neighborhood of \(I^m\) where \(I= [0,1 ]\). For \(f\in \mathbb{R}\{ X_1, \dots, X_m\}\), let \(\widetilde{f}\) be the function which agrees with \(f\) on \(I^m\), and takes the value 0 otherwise. The authors consider the elementary theory \(T_{an}\) of the ordered field \(\mathbb{R}\), enriched with all such \(\widetilde {f}\)’s (which they call restricted analytic functions). In [Ann. Math., II. Ser. 128, No. 1, 79-138 (1988; Zbl 0693.14012)] and [Bull. Am. Math. Soc., New Ser. 15, 189-193 (1986; Zbl 0612.03008)], it was shown that \(T_{an}\) admits quantifier elimination and is \(o\)- minimal.

In this paper, a complete axiomatization for \(T_{an}\) is given. This is used to show that generalized power series fields carry a natural restricted analytic structure, which makes them into models of \(T_{an}\). Embeddings into these models, together with the \(o\)- minimality of \(T_{an}\), are used to establish a crucial valuation theoretic result about models of \(T_{an}\): Suppose \(M\subset N\) are models of \(T_{an}\), and \(y\in N\setminus M\). Let \(M\langle y\rangle\) be the definable closure of \(M\cup\{y\}\) in \(N\). Then the value group (according to the natural valuation on the ordered field \(N\)) of \(M\langle y\rangle\) is the divisible hull of the value group of the subfield \(M(y)\).

Next, the elementary theory \(\text{Th} (\mathbb{R}_{an}, \exp)\) of the ordered field \(\mathbb{R}\), with restricted analytic functions, and unrestricted exponential function, is studied. Consider the theory \(T_{an} (\exp,\log)\): it is \(T_{an}\) augmented with the axioms stating that exp is an order preserving isomorphism from the ordered additive group of the field onto its ordered multiplicative group of positive elements, that exp grows faster than polynomials, and that log is the compositional inverse of exp.

The main result of the paper establishes that \(T_{an} (\exp,\log)\) admits quantifier elimination, has a universal axiomatization, and is complete (and \(T_{an} (\exp)\) is a complete axiomatization of \(\text{Th} (\mathbb{R}_{an},\exp)\)). The quantifier elimination for \(T_{an} (\exp,\log)\) is established by proving an embedding lemma for its models. It is at this point that the key valuation theoretic result mentioned above plays a crucial role. In the last section, the quantifier elimination results are used to show that \((\mathbb{R}_{an},\exp)\) is \(o\)- minimal.

In this paper, a complete axiomatization for \(T_{an}\) is given. This is used to show that generalized power series fields carry a natural restricted analytic structure, which makes them into models of \(T_{an}\). Embeddings into these models, together with the \(o\)- minimality of \(T_{an}\), are used to establish a crucial valuation theoretic result about models of \(T_{an}\): Suppose \(M\subset N\) are models of \(T_{an}\), and \(y\in N\setminus M\). Let \(M\langle y\rangle\) be the definable closure of \(M\cup\{y\}\) in \(N\). Then the value group (according to the natural valuation on the ordered field \(N\)) of \(M\langle y\rangle\) is the divisible hull of the value group of the subfield \(M(y)\).

Next, the elementary theory \(\text{Th} (\mathbb{R}_{an}, \exp)\) of the ordered field \(\mathbb{R}\), with restricted analytic functions, and unrestricted exponential function, is studied. Consider the theory \(T_{an} (\exp,\log)\): it is \(T_{an}\) augmented with the axioms stating that exp is an order preserving isomorphism from the ordered additive group of the field onto its ordered multiplicative group of positive elements, that exp grows faster than polynomials, and that log is the compositional inverse of exp.

The main result of the paper establishes that \(T_{an} (\exp,\log)\) admits quantifier elimination, has a universal axiomatization, and is complete (and \(T_{an} (\exp)\) is a complete axiomatization of \(\text{Th} (\mathbb{R}_{an},\exp)\)). The quantifier elimination for \(T_{an} (\exp,\log)\) is established by proving an embedding lemma for its models. It is at this point that the key valuation theoretic result mentioned above plays a crucial role. In the last section, the quantifier elimination results are used to show that \((\mathbb{R}_{an},\exp)\) is \(o\)- minimal.

Reviewer: S.Kuhlmann (Heidelberg)