A TALK ON THE MANN PROPERTY AND MANN PAIRS; MOSTLY MANN PAIRS

AYHAN GÜNAYDIN

1. Mann Property

Throughout Ω is a big ambient algebraically closed field (of characteristic 0). We are mostly interested in the subgroups Γ of Ω^{×}; later we shall include a subfield of Ω to the game.

Let’s start by giving the main definition right away (after a couple of notations). Let a_{1}, …, a_{n} ∈ Ω, n ≥ 1, and consider the equation

(∗) a_{1}x_{1} + ⋯ + a_{n}x_{n} = 1.

A solution of (∗) is a tuple (s_{1}, …, s_{n}) ∈ Ω^{n} such that a_{1}s_{1} +⋯+a_{n}s_{n} = 1; such a solution is said to be nondegenerate if ∑_{i∈I} a_{i}s_{i} ≠ 0 for all nonempty I ⊆ {1, …, n}, and is said to be in Γ if (s_{1}, …, s_{n}) ∈ Γ^{n}.

We say that Γ has the Mann property if for every a_{1}, …, a_{n} ∈ ℚ^{×} the equation (∗) has only finitely many nondegenerate solutions in Γ. Well, actually it is not really necessary to limit ourselves to rational coefficients, although it takes some algebra to prove that. In any case, from now on I assume it.

The main examples of this property comes from number theory. The reason of the naming like this is that H. Mann proved that the (multiplicative) group of complex roots of unity 𝕌 has the Mann property, using very elementary techniques. Much later than that result some number theorists proved that every multiplicative subgroup of a field of characteristic 0 of finite rank has the Mann property. (Note that 𝕌 is of rank 0.) Another (not finite rank) example is exp(ℚ^{ac}) the exponentials of algebraic numbers (this is proved using the Lindemann’s Theorem, but I’ll mention that in the ‘Mann pairs’ case).

One important feature of the Mann property that it is equivalent to the Mordell-Lang property in the algebraically closed case. More precisely:

Proposition 1.1. For algebraically closed K containing Γ, the following are equivalent:

(1) Γ has the Mann Property;

(2) for every algebraic set V ⊆ K^{n} its trace V ∩ Γ^{n} is a finite union of cosets of subgroups of Γ^{n}; (3) for every X ⊆ K^{n} that is definable in (K, Γ) its trace X ∩ Γ^{n} is

definable in the group Γ.

Other features of the Mann property is that the theory of (Ω, Γ) is axiomatized by the theory of the abelian group Γ and that (Ω, Γ) is (κ-, super-)stable iff the abelian group Γ is (κ-, super-)stable.

2. Mann pairs

Now I will give a stronger form of the Mann property, which is uniform in coefficients. For this fix a subfield k of Ω, which will provide the coefficients.

Call (k, Γ) a Mann pair if for each n ≥ 1 there is a finite Γ(n) ⊆ Γ^{n} such that for all a_{1}, …, a_{n} ∈ k^{×} all non-degenerate solutions of (∗) in Γ lie in Γ(n).

We realized this property when looking for a function field analog for the Mann property, but then realized that H. Mann really proved that (ℚ, 𝕌) is a Mann pair. I am aiming to show the analogous result for finite rank subgroups of characteristic zero fields.

Theorem 2.1. Suppose Ω has characteristic zero, k is algebraically closed, k^{×} ∩ Γ = {1}, and Γ has finite rank. Then (k, Γ) is a Mann pair.

One other thing I want to prove is the following analog of Proposition 1.1.

Theorem 2.2. The following are equivalent:

(1) (k, Γ) is a Mann pair;

(2) for each n ≥ 1 the set {(x, y) ∈ k^{n} × Γ^{n} : x_{1}y_{1} + ⋯ + x_{n}y_{n} = 0} is a finite union of sets X × Y where X ⊆ k^{n} is definable in the field k and Y ⊆ Γ^{n} is definable in the group Γ;

(3) for all m, n, every subset of k^{m} × Γ^{n} definable in (Ω, k, Γ) is a finite union of sets X ×Y with X ⊆ k^{m} definable in the field k and Y ⊆ Γ^{n} definable in the group Γ.

3. Some examples of Mann pairs

It will be useful to consider also homogeneous linear equations.

The equivalence of (1) and (2) in the next lemma expresses being a Mann pair in terms of homogeneous linear equations. Assuming k^{×}∩Γ is finite, the equivalence between (1) and (3) expresses (k, Γ) being a Mann pair purely in terms of the subgroup G = k^{×}Γ of Ω^{×} generated by k^{×} and Γ.

Lemma 3.1. Let G = k^{×}Γ. Then the following are equivalent:

(1) (k, Γ) is a Mann pair;

(2) for each n ≥ 1 there is a finite Γ(n) ⊆ Γ^{n} such that each nondegenerate solution in Γ of each equation a_{1}x_{1} + ⋯ + a_{n}x_{n} = 0 with a_{1}, …, a_{n} ∈ k^{×} lies in αΓ(n) for some α ∈ Γ; (3) k^{×} ∩ Γ is finite, and for each n ≥ 1 there is a finite G(n) ⊆ G^{n} such

that each non-degenerate solution in G of the equation

x_{1} + ⋯ + x_{n} = 1

equals (c_{1}g_{1}, …, c_{n}g_{n}) with c_{1}, …, c_{n} ∈ k^{×} and (g_{1}, …, g_{n}) ∈ G(n).

Corollary 3.2. Suppose (k, Γ) is a Mann pair, and Γ′ is a subgroup of k^{×}Γ such that k^{×} ∩ Γ′ is finite. Then (k, Γ′) is a Mann pair.

3.1. Easy Mann pairs. These are provided by the next lemma.

Lemma 3.3. Suppose Γ is torsion-free. Then the following are equivalent:

(1) for all n ≥ 1 and a_{1}, …, a_{n} ∈ k^{×}, the equation a_{1}x_{1} +⋯+a_{n}x_{n} = 1 has no non-degenerate solution in Γ that is different from (1, …, 1);

(2) whenever n ≥ 1 and γ_{1}, …, γ_{n} ∈ Γ are multiplicatively independent,

then they are algebraically independent over k. If these conditions are satisfied, then (k, Γ) is a Mann pair.

Since π ∉ ℚ^{ac}, the group exp(ℚ^{ac}) ⊆ ℂ^{×} is torsion-free, and by Lindemann’s theorem on exponentials, condition (2) of the lemma is satisfied with k = ℚ^{ac} ⊆ ℂ and Γ = exp(ℚ^{ac}). Thus (ℚ^{ac}, exp(ℚ^{ac})) is a Mann pair. A group with the Mann property can produce a Mann pair in an elementary extension as follows:

Corollary 3.4. Suppose G is a subgroup of k^{×} with the Mann property, (k^{∗}, G^{∗}) is an elementary extension of (k, G), and Γ is a subgroup of G^{∗} such that k^{×} ∩ Γ = {1}. Then (k, Γ) is a Mann pair.

If k has positive characteristic, then Lemma 3.3 is the only source of Mann pairs (k, Γ) with torsion-free Γ:

Corollary 3.5. Suppose k has positive characteristic, Γ is torsion-free, and (k, Γ) is a Mann pair. Then conditions (1) and (2) of Lemma 3.3 are satisfied.

3.2. Notations. In this section we let a⃗ = (a_{1}, …, a_{n}) and b⃗ = (b_{1}, …, b_{n}) range over k^{n}, and α, β, γ over Γ, and

α⃗ = (α_{1}, …, α_{n}), β⃗ = (β_{1}, …, β_{n}), γ⃗ = (γ_{1}, …, γ_{n})

over Γ^{n}. Let n ≥ 1 and put

Σ_{n}(k, Γ) : = {(a⃗, γ⃗) ∈ k^{n} × Γ^{n} : a_{1}γ_{1} + ⋯ + a_{n}γ_{n} = 0},
Σ_{n}(k, Γ; γ⃗) : = {a⃗ ∈ k^{n} : (a⃗, γ⃗) ∈ Σ_{n}(k, Γ)}.

Imposing non-degeneracy yields the set Σ_{n}^{nd}(k, Γ) of all (a⃗, γ⃗) ∈ (k^{×})^{n} × Γ^{n} such that γ⃗ is a non-degenerate solution of a_{1}x_{1} + ⋯ + a_{n}x_{n} = 0. We also introduce for each γ⃗ the corresponding section

Σ_{n}^{nd}(k, Γ; γ⃗) := {a⃗ ∈ (k^{×})^{n} : (a⃗, γ⃗) ∈ Σ_{n}^{nd}(k, Γ)}.

If (a⃗, γ⃗) ∈ Σ_{n}^{nd}(k, Γ), then (a⃗, αγ⃗) ∈ Σ_{n}^{nd}(k, Γ), so Σ_{n}^{nd}(k, Γ) is a union of sets of the form Σ_{n}^{nd}(k, Γ; γ⃗) × Γγ⃗. Being a Mann pair can now be expressed as follows:

(k, Γ) is a Mann pair iff for each n ≥ 1 there is a finite Γ(n) ⊆ Γ^{n} such that

Σ_{n}^{nd}(k, Γ) = ⋃_{γ⃗∈Γ(n)} Σ_{n}^{nd}(k, Γ; γ⃗) × Γγ⃗.

Whenever (k, Γ) is a Mann pair we let Γ(n) for n ≥ 1 be as above.

A close look at these sets give the following.

Lemma 3.6. Suppose (k, Γ) is a Mann pair and n ≥ 1. Then Σ(k, Γ) is a finite union of sets P ×Q, where P is a k-linear subspace of k^{n}, and Q ⊆ Γ^{n} is defined in Γ by a conjunction of atoms in the language of Γ-sets.

4. Mann pairs in function fields of one variable

Let k be algebraically closed, and let F ⊆ Ω be a function field of one variable over k, that is, F is a field extension of finite degree of k(t) for some t ∈ Ω \ k; in particular, F has transcendence degree 1 over k. Below we use some standard facts about such function fields; for proofs of these facts, see Chapter I, §2 of [3].

Let R(F |k), the Riemann space of F over k, be the set of all valuations v : F^{×} → ℤ on F with value group v(F^{×}) = ℤ that are trivial on k. We let v range over R(F |k). For each f ∈ F^{×} we have v(f) ≠ 0 for only finitely many v, and ∑_{v} v(f) = 0. Let D(F |k) be the group of divisors of F over k, that is,

D(F |k) := ⨁_{v} ℤv

is the free abelian group on the Riemann space of F over k. To f ∈ F^{×} we assign its principal divisor (f) := ∑_{v} v(f)v ∈ D(F |k). The group morphism

F^{×} → D(F |k), f ↦ (f)

has kernel k^{×}. In particular, if Γ is a subgroup of F^{×} with k^{×}∩Γ = {1}, then this morphism is injective on Γ, and so the image of Γ under this morphism is an isomorphic copy of Γ inside the free abelian group D(F |k). It follows that each such Γ is free as an abelian group.

Given finite S ⊆ R(F |k), an S-unit is an element u ∈ F^{×} such that v(u) = 0 for all v ∉ S.

Lemma 4.1. Let L be a finite-dimensional k-linear subspace of F and let Γ ⊆ F^{×} be finitely generated with k^{×} ∩ Γ = {1}. Then L ∩ Γ is finite.

Proof. Let b_{1}, …, b_{m} be a basis of the k-linear space L, and let γ_{1}, …, γ_{n} generate the group Γ. Take a finite S ⊆ R(F |k) such that all b_{i} and γ_{j} are S-units. Take a natural number d such that v(b_{i}) ≥ −d for all v ∈ S and i = 1, …, m. Then v(f) ≥ 0 for all f ∈ L and v outside S, and v(f) ≥ −d for all f ∈ L and v ∈ S.

Suppose now that γ ∈ L∩Γ. Then v(γ) = 0 for v outside S and v(γ) ≥ −d for v ∈ S. In view of ∑_{v} v(γ) = 0, this gives v(γ) ≤ |S|d for all v ∈ S. It follows that the image of L∩Γ in D(F |k) is finite, and thus L∩Γ is finite. □

Let u_{1}, …, u_{n} ∈ F not be all zero. We define their height by

H(u_{1}, …, u_{n}) := − ∑_{v} min{v(u_{1}), …, v(u_{n})}.

This height is projective: H(fu_{1}, …, fu_{n}) = H(u_{1}, …, u_{n}) for f ∈ F^{×}.

Example. Let F = k(t) with t transcendental over k. Suppose that the polynomials u_{1}, …, u_{n} ∈ k[t] have no common zero in k, n ≥ 1. It is easy to check that then H(u_{1}, …, u_{n}) = max{deg_{t} u_{1}, …, deg_{t} u_{n}}.

The following important bound is from [2]:

Let Ω have characteristic zero, let g be the genus of the function field F |k, and let S be a finite subset of R(F |k) and n ≥ 2. Suppose u_{1}, …, u_{n} are S-units and (u_{1}, …, u_{n}) is a non-degenerate solution of x_{1} + ⋯ + x_{n} = 0. Then

H(u_{1}, …, u_{n}) ≤ \frac{1}{2}(n − 1)(n − 2)\{|S| + max(0, 2 g −2)\}.

In combination with the previous lemma this has the following consequence:

Corollary 4.2. Suppose Ω has characteristic zero and Γ ⊆ F^{×} is finitely generated with k^{×} ∩ Γ = {1}. Then (k, Γ) is a Mann pair.

Proof. Take a finite S ⊆ R(F |k) such that all γ ∈ Γ are S-units. Let n ≥ 2 and let a_{1}, …, a_{n} ∈ k^{×} and γ_{1}, …, γ_{n} ∈ Γ be such that (γ_{1}, …, γ_{n}) is a non-degenerate solution of a_{1}x_{1} +⋯+a_{n}x_{n} = 0. Dividing by γ_{n} we arrange γ_{n} = 1, and we need to show that this leaves only finitely many possibilities for (γ_{1}, …, γ_{n−1}). Now by the bound above we have v(γ_{i}) ≥ −N for all v ∈ S and i = 1, …, n−1, where N := \frac{1}{2}(n−1)(n−2)\{|S|+max(0, 2 g −2)\}, so each γ_{i} lies in the k-linear subspace

L := {f ∈ F : v(f) ≥ 0 for all v ∉ S, v(f) ≥ −N for all v ∈ S}

of F. Now L is finite-dimensional by [3], p. 7. In view of Lemma 4.1 this gives the desired finiteness. □

Example. To illustrate the effective nature of this proof, assume that Ω has characteristic zero, and consider the case F = k(t) of a rational function field (so g = 0), where Γ is generated as a group by

t − c_{1}, …, t − c_{M}, with distinct c_{1}, …, c_{M} ∈ k.

Let n ≥ 2 be given. Call a tuple (γ_{1}, …, γ_{n}) ∈ Γ^{n} reduced if all γ_{i} lie in k[t] and γ_{1}, …, γ_{n} have no common zero in k when viewed as polynomials in t over k. Let a_{1}, …, a_{n} ∈ k^{×}. Any non-degenerate solution in Γ of the equation

a_{1}x_{1} + ⋯ + a_{n}x_{n} = 0

can be multiplied by an element of Γ to give a non-degenerate reduced solution (γ_{1}, …, γ_{n}). In this situation the inequality from [2] yields

max\{deg_{t} γ_{1}, …, deg_{t} γ_{n}\} ≤ \frac{1}{2}(n − 1)(n − 2)(M + 1),

which is satisfied by only finitely many reduced tuples (γ_{1}, …, γ_{n}), which we can list explicitly. Given a reduced tuple (γ_{1}, …, γ_{n}) satisfying the inequality, the existence of a_{1}, …, a_{n} ∈ k^{×} such that (γ_{1}, …, γ_{n}) is a nondegenerate solution of a_{1}x_{1} + ⋯ + a_{n}x_{n} = 0 is equivalent (effectively) to the existence of a solution to a certain finite system of linear equations and inequations with coefficients in the field ℚ(c_{1}, …, c_{M}).

5. Model theory of Mann pairs

Let L be the language of rings augmented by two distinct unary relation symbols. Let T be the L-theory whose models are the structures (Ω, k, Γ) where Ω is an algebraically closed field with a subfield k, and a subgroup Γ of Ω^{×}. Let L_{Σ}^{f,gr} be the 2-sorted language, with sorts f, gr (the field sort and the group sort), and with the following nonlogical symbols:

• constant symbols 0 and 1 of sort f,

• a unary function symbol − of sort (f; f),

• binary function symbols + and · of sort (f, f; f),

• a constant symbol 1 of sort gr,

• a unary function symbol ^{−1} of sort (gr; gr)

• a binary function symbol · of sort (gr, gr; gr),

• for each n ≥ 1, a 2n-ary relation symbol Σ_{n} of sort (f, …, f, gr, …, gr) (n places of sort f and n places of sort gr).

Of course, “1 used as a symbol of sort f” is different from “1 used as a symbol of sort gr” and likewise with the multiplication symbol. For a model (Ω, k, Γ) of T we construe (k, Γ) as an L_{Σ}^{f,gr}-structure by interpreting the symbols in the obvious way; in particular, each Σ_{n} is interpreted as the previously defined Σ_{n}(k, Γ) ⊆ k^{n} × Γ^{n}.

In the next result we do not assume that (k, Γ) is a Mann pair.

Lemma 5.1. Let (Ω, k, Γ) ⊧ T. Then every subset of k^{m} × Γ^{n} definable in (Ω, k, Γ) is definable in the L_{Σ}^{f,gr}-structure (k, Γ).

Proof. Take an |Ω|^{+}-saturated elementary extension (Ω′, k′, Γ′) of (Ω, k, Γ). It is easy to check that then k′(Γ′) and Ω are linearly disjoint over k(Γ):

Ω′ k′(Γ′)  Ω k′  k(Γ) k

Let a⃗, b⃗ ∈ (k′)^{m} and α⃗, β⃗ ∈ (Γ′)^{n} be such that

tp_{(k′,Γ′)} ((a⃗, α⃗)|(k, Γ)) = tp_{(k′,Γ′)} ((b⃗, β⃗)|(k, Γ)).

It suffices to prove that then

tp_{(Ω′,k′,Γ′)} ((a⃗, α⃗)|(Ω, k, Γ)) = tp_{(Ω′,k′,Γ′)} ((b⃗, β⃗)|(Ω, k, Γ)).

The assumption on (a⃗, α⃗) and (b⃗, β⃗) gives an automorphism of (k′, Γ′) over (k, Γ) that takes (a⃗, α⃗) to (b⃗, β⃗). This automorphism preserves the Σrelations, so it extends to a field automorphism of k′(Γ′) over k(Γ), which extends further to a field automorphism of Ω′ over Ω by linear disjointness of k′(Γ′) and Ω over k(Γ). This yields the desired equality of types. □

The implication (3) ⇒ (2) of Theorem 2.2 is trivial and (2) ⇒ (1) is not that trivial but can be done easily. So this theorem will be established once we prove the part (1) ⇒ (3), which is the next result.

Proposition 5.2. Suppose (Ω, k, Γ) ⊧ T and (k, Γ) is a Mann pair. Then every subset of k^{m} × Γ^{n} that is definable in (Ω, k, Γ) is a finite union of sets X × Y with X ⊆ k^{m} definable in the field k and Y ⊆ Γ^{n} definable in the group Γ.

Proof. As in the proof of the previous lemma we take an |Ω|^{+}-saturated elementary extension (Ω′, k′, Γ′) of (Ω, k, Γ) and tuples a⃗, b⃗ ∈ (k′)^{m} and α⃗, β⃗ ∈ (Γ′)^{n} such that

tp_{k′}(a⃗|k) = tp_{k′}(b⃗|k) and tp_{Γ′}(α⃗|Γ) = tp_{Γ′}(β⃗|Γ).

By Lemma 5.1 it is enough to show that then

tp_{(k′,Γ′)} ((a⃗, α⃗)|(k, Γ)) = tp_{(k′,Γ′)} ((b⃗, β⃗)|(k, Γ)).

The assumption on a⃗ and b⃗ gives an automorphism σ of k′ over k such that σ(a⃗) = b⃗, and the assumption on α⃗ and β⃗ gives an automorphism ϕ of Γ′ over Γ such that ϕ(α⃗) = β⃗. It remains to show that this gives an automorphism (σ, ϕ) of the L_{Σ}^{f,gr}-structure (k′, Γ′). This in turn reduces to establishing the following: Let N be a positive integer and c⃗ ′ ∈ (k′)^{N} and γ′ ∈ (Γ′)^{N}. Then

(k′, Γ′) ⊧ Σ_{N}(c⃗ ′, γ′) ⟺ (k′, Γ′) ⊧ Σ_{N}(σ(c⃗ ′), ϕ(γ′)).

We prove the forward implication. (The backward implication follows in the same way.) Assume (k′, Γ′) ⊧ Σ_{N}(c⃗ ′, γ′). We have Σ_{N}(k, Γ) = ⋃_{i∈I} P_{i} ×Q_{i} where I is finite, each P_{i} ⊆ k^{N} is definable in the field k and each Q_{i} ⊆ Γ^{N} is definable in the group Γ. Then Σ_{N}(k′, Γ′) = ⋃_{i∈I} P′_{i} × Q′_{i} where P′_{i} ⊆ (k′)^{N} is defined in k′ by any formula with parameters from k that defines P_{i} in the field k, and Q′_{i} ⊆ (Γ′)^{N} is defined in Γ′ by any formula with parameters from Γ that defines Q_{i} in the group Γ. Take i ∈ I such that c⃗′ ∈ P′_{i} and γ′ ∈ Q′_{i}. It is clear that then σ(c⃗ ′) ∈ P′_{i} and ϕ(γ′) ∈ Q′_{i}, so (k′, Γ′) ⊧ Σ_{N}(σ(c⃗ ′), ϕ(γ′)), as desired. □

Proposition 5.3. Let (Ω, k, Γ) ⊧ T be such that k is algebraically closed and (k, Γ) is a Mann pair. Then (Ω, k, Γ) is stable, and if Γ is divisible, then (Ω, k, Γ) is ω-stable.

Proof. Take an infinite cardinal κ such that the abelian group Γ is κ-stable. We show that then (Ω, k, Γ) is κ-stable. We can assume k ≠ Ω, and that |Ω| = κ. Take a κ^{+}-saturated elementary extension (Ω′, k′, Γ′) of (Ω, k, Γ).

By the proofs of Lemma 5.1 and Proposition 5.2, the type of an element of k′ over Ω in (Ω′, k′, Γ′) is determined by its type over k in the field k′. Likewise, the type of an element of Γ′ over Ω in (Ω′, k′, Γ′) is determined by its type over Γ in the group Γ′.

Now let t ∈ Ω(k′ ∪ Γ′)^{ac}, say t ∈ Ω(a⃗, γ⃗)^{ac} with a⃗ ∈ (k′)^{m} and γ⃗ ∈ (Γ′)^{n}. Then tp_{(Ω′,k′,Γ′)}(t|Ω) is determined by tp_{k′}(a⃗|k), tp_{Γ′}(γ⃗|Γ) and the specification of a polynomial P (X, Y, T) ∈ Ω[X, Y, T] where X = (X_{1}, …, X_{m}), Y = (Y_{1}, …, Y_{n}) and T is a single indeterminate such that P (a⃗, γ⃗, T) ∈ Ω(a⃗, γ⃗)[T] is irreducible and P (a⃗, γ⃗, t) = 0.

It is also easy to see that all elements of Ω′ outside Ω(k′ ∪ Γ′)^{ac} realize the same type in (Ω′, k′, Γ′) over Ω.

Hence we have at most κ many different 1-types in (Ω′, k′, Γ′) over Ω. □

Remark. One can show that if (Ω, k, Γ) is a model of T and (k, Γ) is a Mann pair, then the subsets k and Γ of Ω are definable in the structure (Ω, k ∪ Γ). Using this fact, Proposition 5.3 also follows from Fact 2.1 and Theorem 4.8 in [1].

5.1. Axiomatizing (Ω, k, Γ). Our goal here is to show that if (Ω, k, Γ) is a model of T and (k, Γ) is a Mann pair with k small in Ω, then the elementary theory of (Ω, k, Γ) is completely determined by the elementary theories of the field k and of the group Γ. We achieve this after adding names for enough elements of k and Γ to witness that (k, Γ) is a Mann pair.

Fix a model (Ω_{0}, k_{0}, Γ_{0}) of T such that (k_{0}, Γ_{0}) is a Mann pair. For each n ≥ 2, fix a finite subset Γ_{0}(n) of Γ_{0}^{n} such that

Σ_{n}^{nd}(k_{0}, Γ_{0}) = ⋃_{γ⃗∈Γ_{0}(n)} Σ_{n}^{nd}(k_{0}, Γ_{0}; γ⃗) × Γ_{0}γ⃗,

and for each γ⃗ ∈ Γ_{0}(n), fix a basis B(γ⃗) ⊆ k_{0}^{n} of the k_{0}-linear subspace Σ_{n}(k_{0}, Γ_{0}; γ⃗) of k_{0}^{n}. Let L(k_{0}, Γ_{0}) be the language L augmented by names for the elements of k_{0} ∪ Γ_{0}.

In the following definition and subsequent remarks (Ω, k, Γ) ranges over models of T that contain (Ω_{0}, k_{0}, Γ_{0}) as a substructure. We construe such (Ω, k, Γ) as an L(k_{0}, Γ_{0})-structure in the obvious way, and say that (Ω, k, Γ) satisfies the Mann-axioms of (Ω_{0}, k_{0}, Γ_{0}) if for each n ≥ 2:

(1) Σ_{n}^{nd}(k, Γ) = ⋃_{γ⃗∈Γ_{0}(n)} Σ_{n}^{nd}(k, Γ; γ⃗) × Γγ⃗;

(2) for each γ⃗ ∈ Γ_{0}(n), the k-linear subspace Σ_{n}(k, Γ; γ⃗) of k^{n} is gener-

ated by B(γ⃗); Remarks. The reason for this terminology is that we have a set Mann(k_{0}, Γ_{0}) of sentences in the language L(k_{0}, Γ_{0}) such that for all (Ω, k, Γ) as above,

(Ω, k, Γ) ⊧ Mann(k_{0}, Γ_{0})
⟺
(Ω, k, Γ) satisfies the Mann axioms of (Ω_{0}, k_{0}, Γ_{0}).

In particular, if (Ω, k, Γ) is an elementary extension of (Ω_{0}, k_{0}, Γ_{0}), then (Ω, k, Γ) satisfies the Mann axioms of (Ω_{0}, k_{0}, Γ_{0}).

Let L^{f} := {0, 1, −, +, ·} be the one-sorted language whose symbols are those of L_{Σ}^{f,gr} involving only sort f, and let L^{f}(k_{0}) be L^{f} augmented by names for the elements of k_{0}. Let L^{gr} := {1, ^{−1}, ·} be the one-sorted language whose symbols are those of L_{Σ}^{f,gr} involving only sort gr, and let L^{gr}(Γ_{0}) be L^{gr} augmented by names for the elements of Γ_{0}. Let n ≥ 1, let x = (x_{1}, …, x_{n}) be a tuple of distinct f-variables, and let y = (y_{1}, …, y_{n}) be a tuple of distinct gr-variables. A careful look at the proof of Lemma 3.6 (which is not included here) for r⃗ = (1, …, 1) yields the following: there are quantifierfree L^{f}(k_{0})-formulas ϕ_{1}(x), …, ϕ_{m}(x) and quantifier-free L^{gr}(Γ_{0})-formulas ψ_{1}(y), …, ψ_{m}(y) such that for all (Ω, k, Γ) ⊧ Mann(k_{0}, Γ_{0}),

Σ_{n}(k^{n}, Γ^{n}) = ⋃_{i=1}^{m} ϕ_{i}(k^{n}) × ψ_{i}(Γ^{n}).

This uniformity is crucial in the proof of the next result.

Theorem 5.4. Let (Ω_{1}, k_{1}, Γ_{1}) and (Ω_{2}, k_{2}, Γ_{2}) be models of T such that (1) [Ω_{1} : k_{1}] > 2 and [Ω_{2} : k_{2}] > 2;

(2) (Ω_{0}, k_{0}, Γ_{0}) ⊆ (Ω_{1}, k_{1}, Γ_{1}) and (Ω_{0}, k_{0}, Γ_{0}) ⊆ (Ω_{2}, k_{2}, Γ_{2});

(3) (Ω_{1}, k_{1}, Γ_{1}) and (Ω_{2}, k_{2}, Γ_{2}) satisfy the Mann-axioms of (Ω_{0}, k_{0}, Γ_{0}). Then: (Ω_{1}, k_{1}, Γ_{1}) ≡_{k_{0}∪Γ_{0}} (Ω_{2}, k_{2}, Γ_{2}) ⟺ k_{1} ≡_{k_{0}} k_{2} and Γ_{1} ≡_{Γ_{0}} Γ_{2}.

For algebraically closed k_{i} this result takes the following form:

Corollary 5.5. Let (Ω_{1}, k_{1}, Γ_{1}) and (Ω_{2}, k_{2}, Γ_{2}) be models of T such that (1) k_{1} and k_{2} are algebraically closed, k_{1} ≠ Ω_{1}, k_{2} ≠ Ω_{2};

(2) (Ω_{0}, k_{0}, Γ_{0}) ⊆ (Ω_{1}, k_{1}, Γ_{1}) and (Ω_{0}, k_{0}, Γ_{0}) ⊆ (Ω_{2}, k_{2}, Γ_{2});

(3) (Ω_{1}, k_{1}, Γ_{1}) and (Ω_{2}, k_{2}, Γ_{2}) satisfy the Mann axioms of (Ω_{0}, k_{0}, Γ_{0}). Then: (Ω_{1}, k_{1}, Γ_{1}) ≡_{k_{0}∪Γ_{0}} (Ω_{2}, k_{2}, Γ_{2}) ⟺ Γ_{1} ≡_{Γ_{0}} Γ_{2}.

References

[1] B. Baizhanov and J.T. Baldwin. Local homogeneity. J. Symbolic Logic, 69:1243–1260, 2004.

[2] W.D. Brownawell and D.W. Masser. Vanishing sums in function fields. Math. Proc. Camb. Phil. Soc., 100:427–434, 1986.

[3] S. Lang. Introduction to Algebraic and Abelian Functions. Graduate Texts in Mathematics, 89. Springer-Verlag, New York-Berlin, second edition, 1982.