On expansions of the real field by complex subgroups

Transcription

1 On expansions of the real field by complex subgroups University of Illinois at Urbana-Champaign July 21, 2016

2 Previous work Expansions of the real field R by subgroups of C have been studied previously. Examples include: (R, Γ), Γ an infinite finite rank subgroup of S 1 : Belegradek and Zilber, The model theory of the field of reals with a subgroup of the unit circle (2008) (R, 2 Z ): van den Dries, The field of reals with a predicate for the powers of two (1985) (R, 2 Z, 2 Z 3 Z ): Günaydın, Model theory of fields with multiplicative groups (2008)

3 Motivation Theorem (Hieronymi, 2010) Let S be an infinite cyclic subgroup of (C, ). Then exactly one of the following holds: 1 Z is definable in (R, S) 2 (R, S) is d-minimal 3 Every open definable set in (R, S) is semialgebraic If S is a finite rank subgroup of S 1, then (R, S) satisfies (3). However, it was not known whether arbitrary finite rank subgroups of C must satisfy one of (1)-(3).

4 In this talk, Γ will be a finite rank subgroup of S 1 which is dense in S 1 and will be a subgroup of R of the form ε Z for some ε > 1. ε 1 1 ε

5 Theorem A Every subset of R m definable in (R, Γ ) is a Boolean combination of sets of the form {x R m : y (Γ ) n s.t. (x, y) W } for some semialgebraic set W R m+2n. Moreover, every open definable set in (R, Γ ) is definable in (R, ). Let Γ = (e iϕ ) Z for some ϕ R \ 2πQ and let = ε Z. From Theorem A, it follows that (R, Γ ) does not satisfy any of (1)-(3).

6 (R, Γ ) does not define Z Let X R be definable in (R, Γ ). By Theorem A, X is a Boolean combination of sets X 1,..., X k, where for i {1,..., k}, X i = {x R : y (Γ ) n i s.t. (x, y) W i } = {x R : (x, y) W i } y (Γ ) n i for some n i 1 and semialgebraic set W i. Each X i is an F σ set, and so X is Borel. But if (R, Γ ) defines Z, then (R, Γ ) defines every projective subset of R.

7 (R, Γ ) is not d-minimal Let P be the projection of Γ onto the real line. By density of (e iϕ ) Z in S 1, P is dense in R. Since Γ is countable, P is codense in R. So (R, Γ ) is not d-minimal.

8 (R, Γ ) is not d-minimal Let P be the projection of Γ onto the real line. By density of (e iϕ ) Z in S 1, P is dense in R. Since Γ is countable, P is codense in R. So (R, Γ ) is not d-minimal. Not every open set definable in (R, Γ ) is semialgebraic The complement of in R >0 is open and definable in (R, Γ ). Moreover, R >0 \ = k Z(ε k, ε k+1 ) so by o-minimality of R, R >0 \ cannot be definable in R. Since both Γ and are definable in (R, Γ ), we now consider (R, Γ, ).

9 The next theorem gives an axiomatization of (R, Γ,, (δ) δ, (γ) γ Γ ). In this theorem, let K be a real closed field. Let G be a dense subgroup of S 1 (K) with γ γ : Γ G a group homomorphism. Let A be a subgroup of K >0 with a group homomorphism δ δ : A such that 1 ε is the smallest element of A greater than 1, and 2 for every k K >0, there is a A such that a k < aε.

10 Theorem B (K, G, A, (δ ) δ, (γ ) γ Γ ) (R, Γ,, (δ) δ, (γ) γ Γ ) if and only if: 1 for every γ Γ and n Z >0, γ is an nth power in Γ if and only if γ is an nth power in G; 2 for all primes p, [p]γ = [p]g; 3 (K, (γ δ ) γ Γ,δ ) satisfies the orientation axioms for Γ ; 4 (K, GA, (γ δ ) γ Γ,δ ) satisfies the Mann axioms for Γ ; 5 all torsion points of G are in Γ. Theorem A is proved using Theorem B.

11 Orientation axioms Definition For n 1, let Q(x 1,..., x n ) Z[x 1,..., x n ] and let γδ := (γ 1 δ 1,..., γ n δ n ) (Γ ) n. The orientation axiom for γδ and Q is the sentence Q(Re(γ 1 δ 1 ),..., Re(γ n δ n )) > 0 if this holds in R, and otherwise it is the sentence Q(Re(γ 1 δ 1 ),..., Re(γ n δ n )) 0. The set of orientation axioms for Γ is the collection of these sentences for each n 1, each Q Z[x 1,..., x n ], and each tuple γδ (Γ ) n.

12 The Mann property Definition Let K be a field of characteristic 0 and G a subgroup of K. For a 1,..., a n Q (n 1), a nondegenerate solution to a 1 x a n x n = 1 ( ) is a tuple (g 1,..., g n ) G n such that a 1 g a n g n = 1 and i I a ig i 0 for each nonempty subset I {1,..., n}. G has the Mann property if every equation of the form ( ) has only finitely many nondegenerate solutions in G.

13 Definition Let L := L or (P, V, Γ, ) be the language of ordered rings, together with a binary relation symbol P, a unary relation symbol V, and names for each δ and γ Γ.

14 Theorem B To prove Theorem B, we prove the following more general result. Theorem B Let M := (K, G, A, (γ) γ Γ, (δ) δ ) and N := (L, H, B, (γ) γ Γ, (δ) δ ) be two models of the L-theory T. Then M N if and only if [p]g = [p]h for all primes p, and for all γ Γ, and n 1, γ is an nth power in G iff γ is an nth power in H.

15 Definition Let H, G be arbitrary groups with H G. H is said to be pure in G if H G [m] = H [m] for all m 1. Definition Let E, F be field extensions of a field k with E, F K for some field K. We say that E and F are free over k if any set S E which is algebraically dependent over F is algebraically dependent over k.

16 Let Sub(K, G, A) be the collection of L or (P, V )-structures (K, G, A ) such that: 1 K is a real closed subfield of K of cardinality less than κ 2 G is a pure subgroup of G containing Γ 3 A is a pure subgroup of A containing 4 K (i) and Q(GA) are free over Q(G A ) 5 For all k (K ) >0, there is a A such that a k < a ε. Let I be the collection of isomorphisms between members of Sub(K, G, A) and Sub(L, H, B) that fix and Γ pointwise.

17 Proof that I is a back-and-forth system Let a K \ K and let ι I. We want to extend ι to ι I such that a dom(ι ). We prove this by considering four cases: 1 a A 2 a Re(G) or a Im(G) 3 a K (Re(GA) Im(GA)) 4 a K \ K (Re(GA) Im(GA))

18 Theorem A: Quantifier reduction Theorem A: Part I Every subset of R m definable in (R, Γ ) is a Boolean combination of sets of the form {x R m : y (Γ ) n s.t. (x, y) W } ( ) for some semialgebraic set W R m+2n. Since Γ and are definable in (R, Γ ), we prove that every subset of R m is definable in (R, Γ, ) is a Boolean combination of sets of the form ( ).

19 To prove this, we prove the following stronger theorem. Theorem Let M := (K, G, A, (γ) γ Γ, (δ) δ ) be a model of T such that [p]g is finite for each prime p. Every subset of K m definable in M is a Boolean combination of subsets of K m defined in M by formulas of the form y z(v (y) P(z) φ(x, y, z)) where φ(x, y, z) is a quantifier free L or (K)-formula. Since Γ is a finite rank subgroup of C, Γ/Γ [n] is finite for each n 1.

20 Next we define an orientation O on S 1 (K). In this picture, O(a, b, c) holds. b a c

21 Definition Let L orm = {O, 1, }. Let Γ G S 1 (K). Let x = (x 1,..., x n ) and let z = (z 11, z 12,..., z n1, z n2 ). For each L orm (Γ)-formula φ(x), there is an L or (Γ)-formula ψ φ (z) such that for all (a 1,..., a n ) G n with a i = (a i1, a i2 ), (G, O, 1, ) = φ(a 1,..., a n ) if and only if (K, <, +,, 0, 1, ) = ψ φ (a 11, a 12,..., a n1, a n2 ). Let Σ orm (Γ) = {ψ φ : φ an L orm (Γ)-formula}.

22 Definition A special L or (P, V )-formula in x is a formula ψ(x) of the form y z(v (y) P(z) θv 1 (y) θ2 P (z) φ(x, y, z)). The following is the main lemma used in proving quantifier reduction. Main lemma Every L-formula ψ(x) is equivalent in T to a Boolean combination of special L-formulas in x.

23 Future work Currently we are studying expansions of R by groups of the form S := (ae iϕ ) Z b Z, where ϕ / 2πQ and a Z b Z is dense in R >0. Let g : a Z b Z (e iϕ ) Z be defined by g(a n b m ) = e iϕn. We consider the structure (R, b Z, a Z b Z, (e iϕ ) Z, g). Question Can we find a theory T and conditions for two models M, N of T to be elementarily equivalent as in Theorem B?

24 Question Let G be a finitely generated subgroup of C and consider (R, G). When can we analyze the definable sets of (R, G) using methods similar to the ones used in Theorem A and Theorem B?