AN UPGRADE FOR “EXPANSIONS OF THE REAL FIELD WITH POWER FUNCTIONS”

CHRIS MILLER

As the title suggests, this brief note is a follow-up to [5] (my first published paper), which the reader is assumed to have at hand. I make more readily available some results from my thesis [6, Chapter IV] that generalize some of the main results from [5], the latter being written just before the technology became available for proving more general results. Though I think these extensions are interesting, the proofs are fairly minor modifications of the material in [5], so I never published them except in my thesis. I also give some new applications, correct a few errors, and give the final data for the references of [5].

Put \overline{ℝ} := (ℝ, <, +, −, ·, 0, 1). Let ℜ be a polynomially bounded o-minimal expansion of \overline{ℝ} having field of exponents K_{0}. Let S ⊆ ℝ and put ℜ^{S} = (ℜ, (x^{s})_{s∈S}). By [8], ℜ^{S} is o-minimal; indeed, so is (ℜ, e^{x}). Unfortunately, the method of proof does not reveal the field of exponents of ℜ^{S}, nor even whether ℜ^{S} is polynomially bounded.^{∗} But the answer is known under some fairly reasonable assumptions.

Let K be the subfield of ℝ generated by S over K_{0}.

Theorem. Suppose that ℜ defines each restriction x^{s}↾[1, 2], s ∈ S. Then ℜ^{S} is polynomially bounded with field of exponents K.

Sketch of proof. For every r ∈ K, x^{r}↾[1, 2] is definable in ℜ^{S}, so we may assume that S = K. By definability of Skolem functions, we may assume that ℜ admits quantifier elimination and is universally axiomatizable. By using [2, Theorem C] instead of [5, 1.2], an easy modification of the proof of [5, 2.5] shows that ℜ^{K} admits quantifier elimination and is explicitly universally axiomatizable over ℜ. (In the proof of [5, 2.4], use K_{0} instead of ℚ, and the reduct of A to the language of ℜ instead of A_{an}.) Without loss of generality, we may assume that ℜ has no relation symbols other than <. Hence, by [1, 5.5, 5.12], ℜ^{K} is o-minimal.^{†} By [1, 5.8], regard the Hardy field ℋ of ℜ^{K} as an elementary extension of ℜ^{K}. In the second and third paragraphs of the proof of [5, 2.4] (again, modified as above), let x be the germ of the identity function on ℝ, A = ℜ^{K} and B = ℋ. Then the resulting structure C is a model of Th(ℜ^{K}) containing ℝ(x) as a Hardy field. On the other hand, by [1, 5.8], ℋ is the smallest model of Th(ℜ^{K}) containing ℝ(x). Hence, C = ℋ, ν(ℋ) = K.ν(x), and K is the field of exponents of ℜ^{K}. □

Corollary 1. Suppose that ℜ defines e^{x}↾[0, 1]. Then ℜ^{S} is o-minimal and polynomially bounded with field of exponents K.

Date: September 10, 2009.

∗This is Speissegger’s opinion [personal communication].

†The proof relies only on Hardy field arguments and is independent of [8].

Proof. For every r ∈ ℝ, the restriction x^{r}↾[1, 2] is definable in (ℜ, e^{x}↾[0, 1]). Apply the theorem. □

Corollary 2 (of the proof). If ℜ is model complete (in a language L extending the language of ordered rings with unity) and defines each partial power x^{s}↾[1, 2], s ∈ S, then ℜ^{S} is model complete (in the obvious extension of L).

Proof. Every r ∈ K is both universally and existentially definable in ℜ^{S} (since K = K_{0}(S)) so it suffices to consider the case that S = K. Since ℜ^{K} admits QE after expanding by the Skolem functions definable in ℜ, and ℜ is model complete, so is ℜ^{K} (without the added Skolem functions). □

Corollary 3. Suppose that ℜ is either the “Gevrey structure” ℝ_{𝒢} defined in [3] or a “Denjoy-Carlemann structure” ℝ_{C(M)} as defined in [7]. Then ℜ^{S} is model complete and polynomially bounded with field of exponents ℚ(S).

Proof. In either case, ℜ is model complete, o-minimal and has field of exponents ℚ. □

Challenge. Many of the results in [5] were reproved, even superceded, without using model theory in [4]. Find “standard” proofs of the above results.

Questions.

— Does (ℜ, e^{x}↾[1, 2]) have field of exponents K_{0}? In every case that we know of, the answer is “Yes”. If not, is it at least polynomially bounded?

— Does (ℜ, (x^{s}↾[1, 2])_{s∈S}) have field of exponents K_{0}? In every case that we know of, the answer is “Yes”. If not, is it at least polynomially bounded?

Corrections to [5].

— Ordered abelian groups should be assumed to have a distinguished positive element 1 > 0, and be regarded in a language extending {<, +, 0, 1}. In particular, this should be reflected in 1.1 and its preceding paragraph.

— In 1.1, replace “A ⊆ B, A′ ⊆ B′” with “A ≼ B, A′ ≼ B′”, and strike “unique”.

— In the Notation preceding 2.2, use L_{an}^{ℚ} and T_{an}^{ℚ} instead of L_{an} and T_{an}.

— In the last line of the proof of 2.2, “u ∈ A” should be “u^{r} ∈ A”.

— In the third line of the proof of 2.4, strike “unique”.

— In the first line of the last paragraph of the proof of 2.4, strike “uniquely”.

— The statement of 3.2 is missing the assumption that T should have QE.

— In the third line on page 89, “α” should be “a”.

Final data for References of [5].

[7] 60 (1995), 74–102. And “pairs” should be “extensions”.

[8] 140 (1994), 183–205.

[9] 85 (1994), 19–56.

[11] 122 (1994), 257–259.

[12] 123 (1995), 2551–2555.

[17] and [18] appeared as one paper: Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Amer. Math. Soc. 9 (1996), 1051–1094.

References

[1] L. van den Dries, A. Macintyre, and D. Marker, The elementary theory of restricted analytic fields with exponentiation, Ann. of Math. (2) 140 (1994), no. 1, 183–205.

[2] L. van den Dries, T -convexity and tame extensions. II, J. Symbolic Logic 62 (1997), no. 1, 14–34.

[3] L. van den Dries and P. Speissegger, The field of reals with multisummable series and the exponential function, Proc. London Math. Soc. (3) 81 (2000), no. 3, 513–565.

[4] J.-M. Lion and J.-P. Rolin, Théorème de préparation pour les fonctions logarithmicoexponentielles, Ann. Inst. Fourier (Grenoble) 47 (1997), no. 3, 859–884 (French, with English and French summaries).

[5] C. Miller, Expansions of the real field with power functions, Ann. Pure Appl. Logic 68 (1994), no. 1, 79–94.

[6] , Polynomially bounded o-minimal structures, Ph.D. thesis, University of Illinois at Urbana-Champaign, 1994.

[7] J.-P. Rolin, P. Speissegger, and A. Wilkie, Quasianalytic Denjoy-Carleman classes and o-min-imality, J. Amer. Math. Soc. 16 (2003), no. 4, 751–777.

[8] P. Speissegger, The Pfaffian closure of an o-minimal structure, J. Reine Angew. Math. 508 (1999), 189–211.

Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, OH 43210

E-mail address: miller@math.ohio-state.edu
URL: http://www.math.ohio-state.edu/~miller