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

Miller, Chris

Ann. Pure Appl. Logic 68, No. 1, 79-94 (1994).

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 \(\mathbb{R}^ K_{\text{an}}\).

Reviewer: A.Prestel (Konstanz)

MSC:

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

Keywords:

axiomatizability; power functions; real-analytic functions

References:

[1]	Bianconi, R., Model completeness results for elliptic and abelian functions, Ann. Pure Appl. Logic, 54, 121-136 (1991) · Zbl 0756.03018
[2]	Bierstone, E.; Milman, P., Semianalytic and subanalytic sets, Inst. Hautes Études Sci. Publ. Math., 67, 5-42 (1988) · Zbl 0674.32002
[3]	Bochnak, J.; Coste, M.; Roy, M.-F., Géométrie algébrique réelle (1987), Springer: Springer Berlin, (in French) · Zbl 0633.14016
[4]	Denef, J.; van den Dries, L., \(P\)-adic and real subanalytic sets, Ann. of Math., 128, 79-138 (1988) · Zbl 0693.14012
[5]	van den Dries, L., A generlization of the Tarski-Seidenberg theorem and some nondefinability results, Bull. Amer. Math. Soc., 15, 189-193 (1986) · Zbl 0612.03008
[6]	van den Dries, L., The elementary theory of restricted elementary functions, J. Symbolic Logic, 53, 796-808 (1988) · Zbl 0698.03023
[10]	Knight, J.; Pillay, A.; Steinhorn, C., Definable sets in ordered structures. II, Trans. Amer. Math. Soc., 295, 593-605 (1986) · Zbl 0662.03024
[13]	Pillay, A.; Steinhorn, C., Definable sets in ordered structures. I, Trans. Amer. Math. Soc., 295, 565-592 (1986) · Zbl 0662.03023
[14]	Ressayre, J.-P., Integer parts of real closed exponential fields, (Clote, P.; Krajicek, J., Arithmetic, Proof Theory and Computational Complexity (1993), Oxford Univ. Press: Oxford Univ. Press New York) · Zbl 0791.03018
[15]	Sacks, G., Saturated Model Theory (1972), W.A. Benjamin: W.A. Benjamin Reading, MA · Zbl 0242.02054
[16]	Tamm, M., Subanalytic sets in the calculus of variations, Acta Math., 146, 167-199 (1981) · Zbl 0478.58010
[17]	Wilkie, A. J., Model completeness results for expansions of the real field I: restricted Pfaffian functions (1992), preprint · Zbl 0892.03013
[18]	Wilkie, A. J., Model completeness results for expansions of the real field II: the exponential function (1992), preprint · Zbl 0892.03013