Complex roots of unity on the real plane

B.Zilber

April-July, 2003 ^{∗}

We consider the theory of the structure

ℂ_{ℝ,roots} = (ℂ, +, ·, R, U)

where R is a unary predicate for the real axis of the complex plane ℂ and U the unary predicate for the complex roots of unity.

A weaker structure (ℂ, +, ·, U) is known to be ω-stable (see [Z]). Anand Pillay asked if ℂ_{ℝ,roots} is well behaving in the context of real model theory, and this has been further discussed in [Mi].

We give here a complete axiomatisation of the theory of ℂ_{ℝ,roots} and prove that the theory allows elimination of quantifiers in a reasonable language. The elimination of quantifiers in fact shows that definable sets in the standard model are finite boolean combinations of some countable unions of semi-algebraic cells.

Let U_{0} = U(ℂ), the complex roots of unity. We use the following Lang property of U_{0} (see [L], Ch.I, s.6 for details):

For every polynomial p(x_{1}, …, x_{n}) over ℚ(U_{0}) (the extension of ℚ by the roots of unity) there is a finite collection of cosets S_{1}, …, S_{k} of algebraic subgroups of (ℂ^{∗})^{n} given by a finite system of equations of the form

x^{m} = γ_{m}

for non-trivial monomials x^{m} and elements γ_{m} ∈ U_{0} such that

∗with corrections suggested by O.Belegradek in January 2005

{a ∈ U_{0}^{n} : p(a) = 0} = ⋃_{i} S_{i} ∩ U_{0}^{n}.

Let T_{ℝ,roots} be the theory of ℂ_{ℝ,roots} and (F, +, ×, R, U) a model of the theory. Since F = R + iR, and R = R(F) is a definable (formal real) subfield, a subset S ⊆ F^{n} can be also vewed as a subset of R^{2n}. Call S ⊆ F^{n} semi-algebraic if S is definable in the field structure (R, +, ×).

Theorem 1 The complete theory T_{ℝ,roots} of ℂ_{ℝ,roots} is given by the following axioms:

1. The universe F has a field structure, F = R + iR with i^{2} = −1 and R a real closed field;

2. U is a multiplicative divisible subgroup of F with n-torsion isomorphic to the cyclic group of order n;

3. for any a ∈ U the unique representation

a = s + it, s, t ∈ R

yields s^{2} + t^{2} = 1;

4. the projection pr U of U on the ’real’ axis R is dense in [−1, 1], that is, for any −1 ≤ a < b ≤ 1 we have (a, b) ∩ pr U ≠ ∅;

5. Lang’s property holds for U, that is

for every polynomial p(x_{1}, …, x_{n}) over ℚ(U_{0}) there is a finite collection of cosets S_{1}, …, S_{k} of algebraic subgroups of (F^{∗})^{n} given by finite system of equations of the form

x^{m} = γ_{m}

for non-trivial monomials x^{m} and elements γ_{m} ∈ U_{0} such that

{a ∈ U^{n} : p(a) = 0} = ⋃_{i} S_{i} ∩ U^{n};

6. R ∖ acl(U) is dense in R, that is for any a < b in R and any semialgebraic function f : (0, 1)^{n} → R (with parameters)

(a, b) ∖ f((pr U)^{n}) ≠ ∅.

The proof of the theorem follows from the lemmas below. We write T for T_{ℝ,roots}.

Lemma 0.1 Let F be a model of T and A a multipicatively independent over U_{0} subset of U(F).

Then A is algebraically independent over U_{0} in the field theoretic sense.

Proof Immediate from Lang’s property. □

Now let F and E be models of T. By Jonsson’s theory there are saturated models F′ ≻ F and E′ ≻ E of some cardinality κ (with 2^{μ} < κ for any μ < κ) we may assume that F′ = F and E′ = E and want to prove that F ≅ E.

Lemma 0.2 Let μ be an ordinal less than κ, A_{μ} ∪ A_{μ}^{∗} be a subset of R(F) A_{μ}^{∗} ⊆ pr U(F), A_{μ} ⊆ R(F), card A_{μ} ≤ |μ|, card A_{μ}^{∗} ≤ |μ|, such that

(i) A_{μ}^{∗} ⊆ pr U(F) is algebraically independent; (ii) A_{μ} is algebraically independent over U(F).

Then (a) pr U(F) ∖ aclA_{μ}^{∗} ≠ ∅

(b) R(F) ∖ acl(U(F) ∪ A_{μ}) ≠ ∅ and

(c) for any a_{μ}^{∗} ∈ pr U(F) ∖ aclA_{μ}^{∗} and a_{μ} ∈ R(F) ∖ acl(U(F) ∪ A_{μ}) the new sets A_{μ+1}^{∗} = A_{μ}^{∗} ∪ {a_{μ}^{∗}} and A_{μ} = A_{μ} ∪ {a_{μ}} satisfy properties (i) and (ii).

Proof (a) follows by saturatedness from the fact that pr U(F) is infinite. (b) follows by saturatedness from axiom 6.

(c) is obvious from assumptions. □

Corollary 1 There exists a transcendence basis of F of the form A∪A^{∗} with A^{∗} ⊆ pr U(F), A ⊆ R(F), pr U(F) ⊆ aclA^{∗}, card A = card A^{∗} = κ.

For X ⊆ R we denote dclX the definable closure of X in the (R, +, ×). We notice that dcl is just the closure under all semi-algebraic functions, that is the functions 0-definable in the formal reals.

We say that a bijection ϕ between subsets A and B of R is a semi-algebraic isomorphism if there is an extension of ϕ to a bijection ϕ : dclA → dclB preserving semi-algebraic functions and < .

Lemma 0.3 There exist transcendence bases Ã ∪ Ã^{∗} ⊆ R(F) and B̃ ∪ B̃^{∗} ⊆ R(E) of F and E correspondingly with Ã^{∗} ⊆ pr U(F), B̃^{∗} ⊆ pr U(E), transcendence bases of pr U and a semi-algebraic isomorphism ϕ : Ã ∪ Ã^{∗} → B̃ ∪ B̃^{∗} such that ϕ(Ã) = B̃.

Proof Let

A ∪ A^{∗} = {a_{i} : i < κ} ∪ {a_{i}^{∗} : i < κ}

and

B ∪ B^{∗} = {b_{i} : i < κ} ∪ {b_{i}^{∗} : i < κ}

be transcendence bases of F and E correspondingly constructed by Corollary 1.

We construct Ã ∪ Ã^{∗}, B̃ ∪ B̃^{∗} and ϕ by transfinite induction.

Suppose that for μ < κ

Ã_{μ} ∪ Ã_{μ}^{∗} = {ã_{i} : i < μ} ∪ {ã_{i}^{∗} : i < μ}

and

B̃_{μ} ∪ B̃_{μ}^{∗} = {b̃_{i} : i < μ} ∪ {b̃_{i}^{∗} : i < μ}

have been constructed with ϕ_{μ} given by the enumeration and

acl(Ã_{μ} ∪ Ã_{μ}^{∗}) ⊇ {a_{i} : i < μ} ∪ {a_{i}^{∗} : i < μ}

and

acl(B̃_{μ} ∪ B̃_{μ}^{∗}) ⊇ {b_{i} : i < μ} ∪ {b_{i}^{∗} : i < μ}.

By back-and-forth method it is enough to show that by letting ã_{μ}^{∗} equal to the first element of A^{∗} ∖ aclÃ_{μ}^{∗} and ã_{μ} equal to the first element of A ∖ acl(pr U(F) ∪ Ã_{μ}) we can extend the isomorphism ϕ to Ã_{μ+1} ∪ Ã_{μ+1}^{∗}.

Denote p_{μ,A}^{∗} the semi-algebraic type of ã_{μ}^{∗} over Ã_{μ} ∪ Ã_{μ}^{∗} and p_{μ,B}^{∗} the corresponding type over B̃_{μ} ∪ B̃_{μ}^{∗}. Notice that the types are given by collections of formulas of the form f_{1} < x < f_{2} for f_{1}, f_{2} semi-algebraic terms over the parameters. Since the parameters of the two types are semi-algebraically conjugated, by elimination of quantifiers in real closed fields, p_{μ,B}^{∗} is consistent. Also the type contains the condition −1 < x < 1 for its variable x. Since the projection pr (U) of U on R is dense in [−1, 1] and p_{μ,B}^{∗}(R) is an intersection of intervals, the type is consistent with x ∈ pr (U). By saturatedness we find b̃_{μ}^{∗} ∈ U(E) realising p_{μ,B}^{∗}.

Now let p_{μ,A} be the semi-algebraic type of ã_{μ} over Ã_{μ+1}^{∗} ∪ Ã_{μ} and p_{μ,B} the correspondent type in E. Again, p_{μ,B}(R) is a given as an intersection of intervals. By axiom 6 this is finitely consistent with the collection of formulas saying that x ∉ acl(U ∪ Ã_{μ+1}^{∗} ∪ Ã_{μ}). Hence by saturatedness there exists

b̃_{μ} ∈ R(F) ∩ p_{μ,B}(F) ∖ acl(U(F) ∪ Ã_{μ+1}^{∗} ∪ Ã_{μ}).

□

Proof of the Theorem. We extend the isomorphism ϕ of the bases of F and E of Lemma 0.3 can be extended to a semi-algebraic isomorphism ϕ̄ : F → E. Now let A^{U} = \{a + i\sqrt{1 − a^{2}} : a ∈ Ã^{∗}\}. Obviously A^{U} ⊆ U(F) and is a maximal algebraically independent subset of U(F). By Lemma 0.1 A^{U} is a maximal multiplicatively independent subset of U(F).

Denote G(A) the group generated by A^{U} and all the roots of any power of elements of A^{U} (the divisible hull of A^{U}. By definitions G(A) = U(F). Define B^{U} and G(B) similarly. ϕ̄ extends uniquely to a semi-algebraic isomorphism A^{U} → B^{U} and hence to U(F) → U(E). Further on we have a unique extension to a semi-algebraic isomorphism ϕ̄ : F → E. Since ϕ̄ preserves U, R and the field structure, we have the required isomorphism between the two models of T.□

We now want to see ℂ_{R,roots} as a structure definable naturally in the reals. Let L^{U} be the definable extension of the language of the field of reals (including <) which contains a name P_{φ,k}(x_{1}, …, x_{m}, y) for each definable predicate on R of the form

∃v_{1}, …, v_{k} ⋀_{i≤k} v_{i} ∈ pr U & φ(x_{1}, …, x_{m}, v_{1}, …, v_{k})

where φ is quantifier-free formula in the language of (R, +, ·, <).

One can see that the predicate U ⊆ R^{2} is definable in L^{U}, so ℝ in the language L^{U} is definable equivalent to the structure ℂ_{R,roots}.

Theorem 2 T_{ℝ,roots} has elimination of quantifiers in the language L^{U}.

Proof

Claim. For any ā, a tuple in a model F of T, the L^{U}-quantifier-free type q =qftp(ā) of the tuple is a complete type.

The theorem follows by compactness from the claim.

To prove the claim we assume that F is saturated and show that if b̄ is a tuple satisfying q in any other saturated model E of the same cardinality, then there is an isomorphism F → E sending ā to b̄.

Up to enumeration we can assume that ā = ⟨a_{1}, …, a_{n}⟩ with {a_{1}, …, a_{l}} algebraically independent over pr U(F) and for each i ∈ {l + 1, …, n}, P_{f_{i},k}(a_{1}, …, a_{l}, a_{i}) holds for some semi-algebraic f_{i} and some k > 0. This effectively means that for some a_{1}^{∗}, …, a_{k}^{∗} ∈ pr U(F) we have a_{i} = f_{i}(a_{1}^{∗}, …, a_{k}^{∗}, a_{1}, …, a_{l}). By choosing k minimal possible we get a_{1}^{∗}, …, a_{k}^{∗} algebraically independent. Since l, the minimal k and corresponding f_{i} are all invariants of type q we have similar {b_{1}, …, b_{l}, b_{1}^{∗}, …, b_{k}^{∗}} for b̄.

Thus we can start the construction of bases by Lemma 0.2 by letting A_{l} = {a_{1}, …, a_{l}}, A_{l}^{∗} = {a_{1}^{∗}, …, a_{k}^{∗}} (assuming l ≥ k) and B_{l} = {b_{1}, …, b_{l}}, B_{l}^{∗} = {b_{1}^{∗}, …, b_{k}^{∗}}. Then the construction of Lemma 0.3 extends A_{l} ∪ A_{l}^{∗} and B_{l} ∪ B_{l}^{∗} to isomorphic bases and we can finish by an isomorphism of the structures as in the proof of Theorem 1. The ismomorphism ϕ : F → E sending ⟨a_{1}^{∗}, …, a_{k}^{∗}, a_{1}, …, a_{l}⟩ to ⟨b_{1}^{∗}, …, b_{k}^{∗}, b_{1}, …, b_{l}⟩ we have thus constructed uniquely determines that ϕ(ā) = ϕ(b̄), since type q tells how the tuple is coordinatised by the basis via semi-algebraic functions. This proves the claim and the theorem. □ References

[L] S.Lang Number Theory III, Springer-Verlag, Berlin-Heidelberg-New York 1991

[Mi] C.Miller Tameness in expansions of the real field , to appear in Logic Colloquium ’01”

[Z] B.Zilber, A note on the model theory of the complex field with roots of unity, www.maths.ox.ac.uk/∼ zilber