Expansions of the real field with power functions. (English) Zbl 0823.03018

The present article extends results of Denef and van den Dries. Let \(\mathbb{R}_{\text{an}}\) denote 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 \([- 1,+ 1]^ m\). It was shown by Denef and van den Dries that this structure is model-complete and o-minimal. The author of the present article extends \(\mathbb{R}_{\text{an}}\) for any subset \(S\subset \mathbb{R}\) to a structure \(\mathbb{R}^ S_{\text{an}}\) by adding for every \(r\in S\) a power function \(p_ r\) defined by \(p_ r(x)= x^ r\) for \(x> 0\) and \(p_ r(x)= 0\) for \(x\leq 0\). It is then shown that for every subfield \(K\) of \(\mathbb{R}\) the theory of \(\mathbb{R}^ K_{\text{an}}\) admits quantifier elimination and is universally axiomatizable. Moreover, the author gives several applications of this result. In particular he studies real-valued functions which are definable in the language of \(\mathbb{R}^ K_{\text{an}}\).


03C60 Model-theoretic algebra
12L12 Model theory of fields
03C10 Quantifier elimination, model completeness, and related topics
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