Imaginaries and invariant types in existentially closed valued differential fields

Definition A.1 (Separated pair).

Let K⊆L be an extension of valued fields. Call a tuple a∈LK-separated if for any tuple λ∈K, val⁡(∑iλi⁢ai)=mini⁡{val⁡(λi⁢ai)}. The pair K⊆L is said to be separated if any finite dimensional sub-K-vector space of L has a K-separated basis.

Proposition A.2.

If K is maximally complete, any extension K⊆L is separated.

Definition A.3 (Uniform stable embeddedness).

Let M be an ℒ-structure and A⊆M. We say that A is uniformly stably embedded if for all formulas ϕ⁢(x;t) there exists a formula χ⁢(x;s) such that for all tuples b∈M there exists a tuple a∈A such that ϕ⁢(A,b)=χ⁢(A,a).

Proposition A.4.

Let M⊧ACVF and ϕ⁢(x;s) an L-formula where x is a tuple of K-variables. There exist an L|Γ-formula ψ⁢(y;u) and polynomials Qi∈Z⁢[X¯,T¯] such that for any N≤M, where the pair K⁢(N)⊆K⁢(M) is separated, and any a∈M, there exist b∈K⁢(N) and c∈Γ⁢(M) such that ϕ⁢(N;a)=ψ⁢(val⁡(Q¯⁢(N,b));c).

Proof.

By elimination of quantifiers (and the fact that 𝐊 is dominant), we may assume that ϕ⁢(x;a) is of the form ψ⁢(val⁡(P¯⁢(x))) where P¯ is a tuple of polynomials from 𝐊⁢(M/X¯), n∈ℕ and ψ is an ℒ|𝚪-formula. Let us write each of the Pi as ∑μai,μ⁢X¯μ. Since the pair 𝐊⁢(N)⊆𝐊⁢(M) is separated, the 𝐊⁢(N)-vector space generated by the ai,μ is generated by a 𝐊⁢(N)-separated tuple d¯∈𝐊⁢(M). Note that |d¯|≤|a¯|. Adding zeros to d¯, we may assume |d¯|=|a¯|. For each i and μ, find λi,μ,j∈𝐊⁢(N) such that ai,μ=∑jλi,μ,j⁢dj. We can rewrite each Pi as ∑jdj⁢Qi,j⁢(X¯,λ¯), where Qi,j∈ℤ⁢[X¯,T¯] does not depend on a¯. For all x∈K⁢(N) we have

The proposition now follow easily by taking b=λ¯ and c=val⁡(d¯).∎

Theorem A.5.

Let K⊆L be a separated pair of valued fields such that L is algebraically closed. Then K is stably embedded in L if and only if Γ⁢(K) is stably embedded in Γ⁢(L), as an ordered Abelian group. Moreover, if Γ⁢(K) is uniformly stably embedded in Γ⁢(L), then K is uniformly stably embedded in L.

Proof.

This follows immediately from Proposition A.4. ∎

Remark A.6.

The computation of Proposition A.4 also applies to the rv map (and the higher order leading terms rvn:𝐊→𝐊/1+n⁢𝔐=𝐑𝐕n in the mixed characteristic case). We get that rvn⁢(Pi⁢(x))=∑jrvn⁢(dj⁢Qi,j⁢(x,λ¯)).

It follows that if the pair K⊆L is separated and L is a characteristic zero Henselian field, K is stably embedded in L if and only if ⋃n𝐑𝐕n⁢(K) is stably embedded in ⋃n𝐑𝐕n⁢(L). If we add angular components (which correspond to splittings of 𝐑𝐕n) and restrict to the unramified case (either residue characteristic zero or positive residue characteristic p and val⁡(p) is minimal positive), then K is stably embedded in L if and only of 𝚪⁢(K) is stably embedded in 𝚪⁢(L) and 𝐤⁢(K) is stably embedded in 𝐤⁢(L).

Corollary A.7.

Let k be any algebraically closed field. The Hahn field K:=k⁢((tR)) is uniformly stably embedded (as a valued field) in any elementary extension.

Proof.

The field K is Henselian, as are all Hahn fields. Its residue field k is algebraically closed and its value group ℝ is divisible. It follows that K is algebraically closed. By Proposition A.2, any extension K⊆L is separated. By Theorem A.5, it suffices to show that ℝ is uniformly stably embedded (as an ordered group) in any elementary extension. But that follows from the fact that (ℝ,<) is complete and (ℝ,+,<) is o-minimal, see [2, Corollary 64]. ∎

Remark A.8.

An easy consequence of this result is that the constant field C𝐊 is stably embedded in models of VDFℰ⁢𝒞. Indeed by quantifier elimination, we only need to show that C𝐊 is stably embedded in 𝐊 as a valued field. But that follows from Corollary A.7 and the fact that for any k⊧DCF0, K=k⁢((tℝ))⊧VDFℰ⁢𝒞 (for the derivation described in Example 2.3) and its constant field CK=Ck⁢((tℝ)) is uniformly stably embedded in K.

It then follows from quantifier elimination that C𝐊 is a pure algebraically closed field.