ZILBER’S DICHOTOMY FOR DIFFERENTIALLY CLOSED
FIELDS WITH AN AUTOMORPHISM
RONALD F. BUSTAMANTE MEDINA
Abstract. The theory of difference-differential fields of characteristic zero has
a model-companion denoted by DCFA. In [2] we prove its main properties. In
[3] we proved a weak version of Zilber’s dichotomy for DCFA. In this paper
we use arc spaces techniques as developed by Moosa, Pillay and Scanlon in
[10] to suppress the extra hypothesis needed in [3] and prove the full Zilber’s
dichotomy for DCFA, we also state how these techniques generalise to partial
differential fields with an automorphism.
1. Introduction and preliminaries
A difference-differential field is a differential field with an automorphism which
commutes with the derivation. For the case of characteristic zero, the theory of
difference-differential has a model-companion that we denote DCFA.
If (K,σ,D) is a model of DCFA, the fixed field of K is Fixσ = {x ∈ K : σ(x) =
x} and the field of constants of K is C = {x ∈ K : Dx = 0}.
Our goal is to prove the following theorem, known as Zilber dichotomy.
Theorem 1. If p is a type of SU -rank 1, then it is either 1-based or non orthogonal
to Fixσ ∩ C.
This dichotomy states, intuitively speaking, that a ”minimal” set is either ”al-
gebraic” or ”geometrically simple”. Analogues for this dichotomy are satisfied in
both differentially closed fields (models of DCF, the model companion or the the-
ory of differential fields) and existentially closed difference fields (models of ACFA,
the model companion of the theory of difference fields). Both proofs ([7], and [5])
depend heavily on Zariski geometries. These results lead to Hrushovsky’s proofs
of the Mordell-Lang conjecture and the Manin-Mumfurd conjecture, respectively.
Pillay and Ziegler proved both dichotomies for the case of characteristic zero [14]
using algebraic jet spaces. A key point in their proofs is the fact that in mod-
els of DCF tp(a/F ) has finite rank if and only if the transcendence degree of the
differential field generated by the tuple a and F over F is finite, where F is a
differential subfield of a the model. This transcendence degree is what we call the
dimension of a over F , denoted dim(a/F ). An analogous statement holds for mod-
els of ACFA.These equivalences allow the authors to replace an element of finite
rank with something interdefinable which, by finite-dimensionality, turns out to be
Date: July, 2020.
2000 Mathematics Subject Classification. 11U09, 12H05, 12H10 .
Key words and phrases. model theory of fields, supersimple theories, difference-differential
fields, Zilber’s dichotomy, partial differential fields.
Research supported by Centro de Investigaciones en Matemática Pura y Aplicada and Escuela
de Matemática, Universidad de Costa Rica.
1
2 RONALD F. BUSTAMANTE MEDINA
a finite sequence. Unfortunately this equivalence does not hold for DCFA. In [4] we
give an example of a type with SU -rank 1, but with infinite dimension. So following
the jet spaces techniques we were able to prove the following weaker dichotomy.
Theorem 2. [3]. Let (U , σ,D) be a saturated model of DCFA and let K = acl(K) ⊂
U . Let a ∈ U such that SU(tp(a/K)) = 1 and tp(a/K) is finite-dimensional. Then
tp(a/K) is either 1-based or non orthogonal to Fixσ ∩ C.
In this paper we want to suppress the finite-dimensional hypothesis. For this
we follow [10] where the authors use algebraic arc spaces to prove a dichotomy for
differentially closed fields with finitely many commuting derivations (DCFn). The
main theorem states that if p is a regular non-locally modular type then there is
a definable subgroup of the additive group whose generic type is regular and non
orthogonal to p.
We want to prove an analogue theorem for our case, namely:
Theorem 3. Let p be a regular non-locally modular type in U . Then there is a
definable subgroup of the additive group whose generic type is regular and non-
orthogonal to p.
How can we link Theorem 2 and Theorem 3 to prove Theorem 1? The next
proposition on definable subgroups of the additive group will do the job:
Proposition 1.1. [4]. Let G be a definable subgroup of Gna .
(1) G has no proper subgroup of finite index.
(2) G is quantifier-free definable.
(3) If H a definable subgroup of G. Then G/H is definably isomorphic to a
subgroup of Gla for some l.
(4) If G has infinite dimension then SU(G) ≥ ω.
Now, provided that Theorem 3 holds, we can easily prove Theorem 1.
Proof of Theorem 1:
Suppose that p is not 1-based, then it is non-locally modular. As SU(p) = 1 it
is regular. By Theorem 3 there is a definable subgroup G of Ga whose generic type
q is regular and non-orthogonal to p.
p 6⊥ q implies that SU(q) = α+ 1 for some α. Then, by 5.4.3 of [15], G contains
a definable subgroup N such that SU(G/N) < ω, and by 1.1 G must be finite-
dimensional.
By Theorem 2, q 6⊥ Fixσ ∩ C and by transitivity p 6⊥ Fixσ ∩ C.
2
DCF is complete, ω-stable and eliminates quantifiers, these facts are key in [10],
on the other hand DCFA is not complete, no completion is stable nor eliminates
quantifiers. However its completions are easily described, they are supersimple,
quantifier-free stable and eliminates imaginaries. So we need to state some facts
that allow us to build a set-up similar to the one exposed in [10]. Section 2 is
devoted to this. In section 3 we give a brief description and list the main theorems
on DCFA. In section 4 we use arc spaces to give a proof of 3. Finally, in section 5
we give details on how the same reasonings apply to partial differential fields with
an automorphism.
ZILBER’S DICHOTOMY FOR DIFFERENTIALLY CLOSED FIELDS WITH AN AUTOMORPHISM3
2. Types in supersimple theories
Throughout this section T shall denote a supersimple theory which eliminates
imaginaries and M a saturated model of T . This implies that there is a good notion
of independence, and thus of forking. Moreover all types are ranked by the SU -
rank. With independence we define orthogonality: if A ⊂M is a set of parameters
and p, q are complete types over A. We say that p and q are orthogonal (p ⊥ q) if
for every B ⊇ A and realisations a of p and b of q, we have that a is independent
of b over B (a |̂ Bb).
Proposition 2.1. ([15], 5.1.12) Let a ∈ M and A ⊂ M . Let us suppose that
SU(a/A) = β + ωα · n, with n > 0 and ωα+1 ≤ β < ∞ or β = 0. Then tp(a/A)
is non-orthogonal to a type of SU -rank ωα. Moreover there is b ∈ acl(Aa) with
SU(b/A) = ωαn.
Definition 2.2. Let p, q ∈ S(A). We say that q is p-internal if for every realization
a of q there is a set B such that B |̂ Aa and a tuple c of realizations of p such that
a ∈ dcl(Bc). A set X definable over A is p-internal if for every tuple a of X,
tp(a/A) is p-internal. If we replace dcl by acl above we say that q (or X) is almost
p-internal.
Definition 2.3. Let p be a (possibly partial) type over A and q = tp(a/B) a type.
The p-weight of q, denoted by wp(q), is the largest integer n such that there are
C ⊃ A ∪ B, a tuple a1, . . . , an of realizations of p which are independent over C,
and a realization b of q such that (a1, . . . , an) |̂ |̂ /|̂AC, b BC and ai Cb for every
i = 1, . . . , n. If p is the partial type x = x we say weight instead of p-weight and it
is denoted by w(q).
Definition 2.4. Let A, B and C be sets. We say that A dominates B over C
if for every set D, D |̂ CA implies D |̂ CB. Let p, q be two types. We say that p
dominates q if there is a set C containing the domains of p and q and realizations a
and b of non-forking extensions of p and q to C respectively, such that a dominates
b over C. We say that p and q are equidominant if p dominates q and q dominates
p.
A type is said to be regular if it is orthogonal to all its forking extensions. For
regular types, equidominance is an equivalence relation and non-orthogonality is
transitive ([15], section 5.2).
Let p and q be two complete types. We say that q is hereditarily orthogonal to
p if every extension of q is orthogonal to p.
Definition 2.5. (1) Let p be a type and A a set. The p-closure of A, clp(A)
is the set of all a such that tp(a/A) is hereditarily orthogonal to p.
(2) A type p is called locally modular if for any A containing the domain of
p, and any tuples a and b of realizations of p, we have a |̂ Cb where C =
clp(Aa) ∩ clp(Ab)
For the dichotomy we talk about 1-basedness, which is related with local mod-
ularity.
Definition 2.6. (1) Let A ⊂M and let S be an (∞)-definable set over A. We
say that S is 1-based if for every m,n ∈ N, and a ∈ Sm, b ∈ Sn, a and b
are independent over acl(Aa) ∩ acl(Ab).
4 RONALD F. BUSTAMANTE MEDINA
(2) A type is 1-based if the set of its realizations is 1-based.
For an important characterization of locally modular types we need to introduce
the following definitions.
Definition 2.7. (1) Let p ∈ S(A) be regular and let q be a type over a set
B ⊃ A. We say that q is p-simple if there is C ⊃ B and a set X of
realizations of p and a realization a of q with a |̂ BC such that tp(a/CX)
is hereditarily orthogonal to p.
(2) Let p be a regular type over A and let q be a p-simple type. We say that q
is p-semi-regular if it is domination equivalent to a non-zero power of p
Remark 2.8. If a, b are tuples of M , the canonical base of a over b denoted by
Cb(a/b) the smallest algebraically closed subset of M over which tp(a/b) does not
fork. As T is simple, canonical bases exist (see [1] for the details). 3.3 of [15]
implies that for B ⊂ M, a tuple a of M , and a sequence ai of realizations of
tp(a/B) independent over B, there is some m for which Cb(a/B) ⊆ dcl(a1 · · · am)
.
Proposition 2.9. ([15],3.5.17) A type p is locally modular if and only if for any
two models M and N with N ≺ M , and any tuple of realizations a of p over M
such that tp(a/N) is p-semi-regular, Cb(a/M) ⊂ clp(Na).
Lemma 2.10. Let A be a subset of M , a a tuple and b ∈ acl(Aa). Let q = tp(a, b/A)
be a regular type and let p = tp(b/A). Then p is locally modular if and only if q is
locally modular.
Proof:
As being hereditarily orthogonal to p is the same as being hereditarily orthogonal
to q, by definition, for any set B, clp(B) = clq(B).
Let (a1, b1) and (a2, b2) be tuples (of tuples) of realizations of q.
Claim: ai ∈ clp(bi) for i = 1, 2:
Proof of the claim: Suppose that a1 6∈ clp(b1). Then there exists a non-forking
extension r of tp(a1) over B such that r is non-orthogonal to tp(b1). Let c and d
be realisations of the partial type r ∪ q. Then tp(c, d/B) is a forking extension of
q. The element c can be chosen dependent over B from a realisation (c′, d′) of a
forking extension of q over B, and this contradicts the regularity of q. The same
argument applies to a2 and clp(b2) and the claim is proved.
The following equation holds:
clq(a1, b1) ∩ clq(a2, b2) = clp(b1) ∩ clp(b2) =def C.
It follows immediately that the local modularity of q implies the local modu-
larity of p. Conversely, assume that p is locally modular. Then b |̂1 Cb2. Let
D = Cb(a1b1/acl(Cb2)). By 2.8 D is contained in the algebraic closure of indepen-
dent realizations of tp(a1b1/acl(Cb2)) this implies that tp(D/C) is almost-internal
to the set of conjugates of tp(a1/Cb1), and is therefore hereditarily orthogonal
to p. Hence D ⊂ cl |̂p(C) = C, and a1b1 Cb2. A similar reasoning gives that
Cb(a2b2/acl(Ca1b1)) ⊂ C.
2
ZILBER’S DICHOTOMY FOR DIFFERENTIALLY CLOSED FIELDS WITH AN AUTOMORPHISM5
Lemma 2.11. Let A = acl(A) a subset of M of T and let a be a tuple in M .
Assume that tp(a/A) has SU-rank β + ωα = β ⊕ ωα, with β > ωα, and has weight
1. Then there is b ∈ acl(Aa) such that SU(b/A) = ωα.
Proof:
By 2.1, there is some C = acl(C) ⊃ A independent from a over A and a tuple c
such that SU(c/C) = ωα, and c and a are not independent over C. Let B be the
algebraic closure of Cb(Cc/acl(Aa)). Then by 2.8 B is contained in the algebraic
closure of finitely many (independent over Aa) realizations of tp(Cc/acl(Aa)), say
C1c1, . . . , Cncn. Let D = acl(C1, . . . , Cn). Then D is independent from a over
A, and each ci is not independent from a over D. Since tp(a/A) has weight 1, so
does tp(a/D), and therefore for each 1 < i ≤ n, c1 and ci are not independent
over D. Thus SU(ci/Dc ) < ω
α
1 , and therefore SU(c1, . . . , cn/D) < ω
α2. As
D is independent from a over A, and B ⊂ acl(D, c1, · · · , cn) ∩ acl(Aa), we get
SU(B/A) < ωα2. Since SU(c/C) = ωα and SU(c/CB) < ωα, then SU(B/C) ≥
ωα, and as B |̂ AC we have SU(B/A) ≥ ωα. By Lascar’s inequalities we have
SU(a/AB) + SU(B/A) ≤ β + ωα. As SU(B/A) ≥ ωα we have that SU(B/A) =
δ + ωα with δ ≥ ωα or δ = 0, and SU(B/A) < ωα2 implies that δ = 0.
2
We end this section with two useful results proved in [15], section 5.2.
Proposition 2.12. A type in a supersimple theory is equidominant with a finite
product of regular types.
Proposition 2.13. If p 6⊥ q and q 6⊥ r then there is a conjugate r′ of r such that
p 6⊥ r′.
3. Difference-differential fields
A difference-differential field is a differential field (K,D) with an automorphism
σ of K which commutes with D.
The theory of difference-differential fields of characteristic zero has a model-
companion which we denote DCFA ([2]). Before we give an axiomatization of
this theory we need to introduce some definitions regarding varieties defined in
differential fields.
Definition 3.1. Let (K,D) be a differential field, and let V ⊂ An be a va-
riety, let F (X) be a finite tuple of polynomials over K generating I(V ) where
X = (X1, · · · , Xn).
(1) We define the first prolongation of V , τ1(V ) by the equations:
F (X) = 0, JF (X)Y
t D
1 + F (X) = 0
where Y1 is an n-tuple, F
D denotes the tuple of polynomials obtained by ap-
plying D to the coefficients of each polynomial of F , and JF (X) is the Jaco-
bian matrix of F (i.e. if F = (F1, · · · , Fk) then JF (X) = (∂Fi/∂Xj)1≤i≤k,1≤j≤n).
(2) For m > 1, we define the m-th prolongation of V by induction on m:
Assume that τm−1(V ) is defined by F (X) = 0, JF (X)Y
t D
1 + F (X) =
0, · · · ,
JF (X)Y
t
m−1 + fm−1(X,Y1, · · · , Ym−2) = 0. Then τm(V ) is defined by:
(X,Y1, · · · , Ym−1) ∈ τm−1(V )
6 RONALD F. BUSTAMANTE MEDINA
and
JF (X)Y
t + JDm F (X)Y
t
m−1 + Jf − (X,Y1, · · · , Ym−2)(Y1, · · · , Y tm 1 m−1)
+fDm−1(X,Y1, · · · , Ym−2) = 0.
(3) Let W ⊂ τm(V ) be a variety. We say that W is in normal form if, for every
i ∈ {0, · · · ,m − 1}, whenever G(X,Y1, · · · , Yi) ∈ I(W ) ∩K[X,Y1, · · · , Yi]
then
JG(X,Y1, · · · , Yi)(Y1, · · · , Yi+1)t +GD(X,Y1, · · · , Yi) ∈ I(W ).
(4) Let W ⊂ τm(V ) be a variety in normal form.
A point a (in some extension of K) is an (m,D)-generic of W over K if
(a,Da, · · · , Dma) is a generic of W over K and for i > m, we have
tr.dg(Dia/K(a, · · · , Di−1a)) = tr.dg(Dma/K(a, · · · , Dm−1a)).
Now we can give an axiomatization for DCFA.
Fact 3.2. [2]. (K,D, σ) is a model of DCFA if
(1) (K,D) is a differentially closed field.
(2) σ is an automorphism of (K,D).
(3) If U, V,W are varieties such that:
(a) U ⊂ V × V σ projects generically onto V and V σ.
(b) W ⊂ τ1(U) projects generically onto U .
(c) π1(W )
σ = π2(W ) (we identify τ1(V × V σ) with τ1(V ) × τ1(V )σ) and
let π1 : τ (V × V σ) → τ (V ) and π : τ (V × V σ) → τ (V )σ1 1 2 1 1 be the
natural projections).
(d) A (1, D)-generic point of W projects onto a (1, D)-generic point of
π1(W ) and onto a (1,D)-generic point of π2(W ).
Then there is a tuple a ∈ V (K), such that (a, σ(a)) ∈ U and
(a,Da, σ(a), σ(Da)) ∈W .
This theory is not complete, but its completions are easily described. It’s proper-
ties in fact are in general very similar to the ones of ACFA: independence is defined
by linear disjointness, the models of DCFA eliminates imaginaries (moreover, they
satisfy the Independence Theorem over algebraically closed sets), the completions
of DCFA are supersimple and thus types are ranked by the SU -rank; forking is
determined by quantifier-free formulas, this implies that DCFA is quantifier-free
ω-stable; in a model of DCFA the difference-differential Zariski topology (defined
in analogy with Zariski topology in algebraically closed fields) is Noetherian. All
these properties are proved in [2].
Since DCFA is quantifier-free ω-stable, we can define canonical bases for quantifier-
free types as in stable theories as in [6]. We denote the canonical basis of the
quantifier-free type of a over K as Cb(qftp(a/K)). It does not coincide with the
canonical base of tp(a) as defined for simple theories. However, as DCFA satisfies
the independence theorem over algebraically closed sets, Cb(p) will be contained in
acl(Cb(qftp(a/K)).
4. Arc spaces in difference differential fields
First we define algebraic arc spaces, for the formal definition we refer to [10].
Let K be a field, and K(m) the K-algebra K[]/(m+1). Then, identifying K(m)
with K · 1 ⊕ K ·  . . . ⊕ K · m, we see that the K-algebra K(m) is quantifier-free
interpretable in K, if we encode elements of K(m) by (m+ 1)-tuples of K.
ZILBER’S DICHOTOMY FOR DIFFERENTIALLY CLOSED FIELDS WITH AN AUTOMORPHISM7
Let V ⊂ A` be a variety defined over K. For m ∈ N, we consider the set V (K(m))
of K(m)-rational points of V .
Using the quantifier-free interpretation of K(m) in K, we may identify V (K(m))
with a subvariety AmV (K) of A(m+1)`(K). The variety AmV is called the m-th
arc bundle of V . More precisely, if f1, . . . , fk ∈ K[X1, . . . , X`] generate the ideal
I(V ), then the ideal of AmV is generated by the polynomials fj,t ∈ K[Xi,t : 1 ≤
i ≤ `, 0 ≤ t ≤ m], 1 ≤ j∑≤ k, 0 ≤ t ≤ m, wh∑ich are defined by the identitym m
f t tj(( xi,t )1≤i≤l) = fj,t((xi,t)1≤i≤l)
t=0 t=0
modulo (m+1)
Ifm = 1 the polynomials definingA1V are determined by fj((xi,0+xi,1)1≤i≤l) =
fj,t((x
2
i,0)1≤i≤l) + fj,t((xi,1)1≤i≤l) modulo () and we identify A1V with the tan-
gent bundle T (V ).
If r > m, the natural map K(r) → K(m) induces a map V (K(r)) → V (K(m)),
which in turn induces a morphism ρr,m : ArV → AmV .
In general, given a morphism of varieties f : U → V defined over K, the natural
morphism U(K(m)) → V (K(m)) induced by f gives rise to a morphism Amf :
AmU → AmV .
Let us write ρm for ρm,0. For a ∈ V (K) the m-th arc space of V at a, AmVa is
the fiber of ρm over a.
We follow [10] to prove our theorem. Our propositions 4.6, 4.7, 4.10, 4.12, 4.14,
4.17, 4.19, 4.20, 4.21, 4.22, 4.24, 4.25 and 4.26 are difference-differential versions of
2.9, 2.10, 2.11, 2.12, 3.1, 3.6, 3.9, 3.10, 3.11, 3.12, 3.15 and 3.16 of [10]. Some of
the proofs are very similar, we give the details where the proofs are different or in
case they give some useful insights.
The following three lemmas are crucial, and allow us to characterise varieties by
their arc spaces.
Lemma 4.1. [10]. Let U, V be two algebraic varieties, and let f : U → V be
a morphism, all defined over K. Let m ∈ N and am ∈ AmU(K) be such that
ā = ρm(a) and f(¯a) = ρm(f(a)) are non-singular. Let U
′ be the fiber of ρm+1,m :
A ′m+1U → AmU over a and V the fiber of ρm+1,m : Am+1V → AmV over Amf(a).
Let ā = ρm(a). Then there are biregular maps ϕ : U
′
U → T (U)ā and ϕ ′V : V →
T (V )f(ā) such that the following diagram is commutative:
′ Am(f)U // V ′
ϕU ϕV
 
dfā
T (U) //ā T (V )f(ā)
Lemma 4.2. [10]. Let U, V be algebraic varieties defined over K, and let f : U → V
be a dominant map defined over K. Let a ∈ U(K) be non-singular such that f(a)
is non-singular and the rank of dfa equals dimV . Then for every m ∈ N the map
Am(f) : AmUa(K)→ AmVf(a)(K) is surjective.
Lemma 4.3. [10]. Let U, V,W be algebraic varieties defined over K such that
U, V ⊂ W . Let a ∈ U(K) ∩ V (K) be non-singular. Then U = V if and only if
AmUa(K) = AmVa(K) for all m ∈ N.
8 RONALD F. BUSTAMANTE MEDINA
Let (U , σ,D) be a saturated model of DCFA, let K be a difference-differential
subfield of U .
Let V be an algebraic variety of the affine space of dimension k over U .
Let ∇m : V → τm(V ) be defined by x 7→ (x,Dx, · · · , Dmx) and let πl,m :
τl(V ) → τm(V ) be the natural projection for l ≥ m. Sm(V ) will denote the
Zariski closure of {(x, · · · , σm(x)) : x ∈ V }. Let qm : V → Sm(V ) be defined by
x 7→ (x, · · · , σm(x)) and let pl,m : Sl(V ) → Sm(V ) be the natural projections for
l ≥ m.
We now define a notion of difference-differential prolongation. Let Φm(V ) =
τm(Sm(V )), let ψm : V → Φm(V ) be such that x 7→ ∇m(qm(x)) and for l ≥ m let
tl,m : Φl(V ) → Φm(V ) be defined by tl,m = πl,m ◦ pl,m. Let us denote πl = πl,0,
p 1l = pl,0, tl = tl,0, Φ(V ) = Φ (V ) = Φ1(V ) and Φ
m+1(V ) = Φ(Φm(V )).
We define ψ = ψ1 = ψ1 : V → Φ(V ) and ψm+1(V ) = ψ(ψm) : V → Φm+1(V ).
We extend σ and D to K(m) by defining σ() =  and D = 0. Since σ and D
commute, ∇mqm(x)∑is a permutation of qm∇mx. For x = (∑x0, · · · , xm) ∈ K(m)m m
we identify it with i=0 xi
i, we can identify Φm(x) with
i
i=0 Φm(xi) Then
we can identify Ar(Sm(V ))(K) with Sm(Ar(V ))(K). We can thus assume that
Ar(Φm(V ))(K) = Φm(Am(V ))(K). All this is done with greater generality in [12]
and [13]
Let X be a (σ,D)-variety given as a (σ,D)-closed subset of an algebraic variety
X̄. We define Φm(X) as the Zariski closure of ψm(X) in Φm(X̄). Thus X is de-
termined by the prolongation sequence {tl,m : Φl(X) → Φm(X)) : l ≥ m}, since
X(U) = {a ∈ X̄(U) : ψl(a) ∈ Φl(X)∀l}. We call this sequence the prolongation
sequence of X.
Proposition 4.4. Let {Vl ⊂ Φl(V̄ ) : l ≥ 0} be a sequence of algebraic varieties
and {tm,l : Vm → Vl,m ≥ l} a sequence of morphisms such that:
(1) tl+1,l  Vl+1 → Vl is dominant.
(2) After embedding Φ (V̄ ) in Φll (V̄ ) and Φl+1(V̄ ) in Φ
l+1(V̄ ),
(a) Vl+1 is a subvariety of Φ(Vl).
(b) Let π′1 : Φ(Vl) → τ(Vl) and π′2 : Φ(Vl) → τ(V σl ) be the projections
induced by Φ(V ) ⊂ τ(V ) × τ(V σl l l ); then π′1(Vl+1)σ and π′2(Vl+1) have
the same Zariski closure.
Then there is a (unique) (σ,D)-variety V with prolongation sequence {tm,l : Vm →
Vl,m ≥ l} .
Proof:
For each l, as the maps πm,j are dominant, the system {pm,l(Vm), πm,j : m >
l
j ≥ l} defines a differential subvariety W σl of V̄ × · · · × V̄ .
Condition (1). implies that form sufficiently large, an (m,D)-generic of pm,l+1(V )
is sent by pl+1,l to an (m,D)-generic of pm,l(V ). Hence, a D-generic of Wl+1 is
sent by pl+1,l to a D-generic of Wl (that is, a generic in the sense of DCF).
By conditions (2) (b) and (1), the map t′l+1,l : Vl+1 → V σl induced by Φ(V )→ V σl l
is dominant. Hence, considering the natural projection p′l+1,l : Sl+1(V̄ )→ Sl(V̄ )σ,
ZILBER’S DICHOTOMY FOR DIFFERENTIALLY CLOSED FIELDS WITH AN AUTOMORPHISM9
and reasoning as above, we obtain that p′l+1,l sends a D-generic of Wl+1 to a D-
generic of Wσl . Hence by 3.2, for every l there is a such that ψl(a) is a generic of Vl
over K. By saturation, there is a such that for all l, ψl(a) is a generic of Vl. Then
{tm,l : Vm → Vl,m ≥ l} is the prolongation sequence of the (σ,D)-locus of a over
K.
2
We define non-singular points in analogy with the corresponding notion in [10].
Definition 4.5. Let X be a (σ,D)-subvariety of the algebraic variety X̄. We say
that a point a ∈ X is non-singular if, for all l, ψl(a) is a non-singular point of
Φl(X), the maps dtl+1,l and dt
′
l+1,l at ψl+1(a) have rank equal to dimXl and the
maps dπ′1 and dπ
′
2 (as defined above) at ψl+1(a) have rank equal to the dimension
of the Zariski closure of π′1(Φl+1(X)).
The next proposition is essentially 4.7 of [13].
Proposition 4.6. Let (K,σ,D) be a model of DCFA. Let V be a (σ,D)-variety
given as a closed subvariety of an algebraic variety V̄ . Let m ∈ N and a ∈ V (K)
a non-singular point. Then {Am(tr,s) : AmΦr(V )ψ → A Φr(a) m s(V )ψ (a), r ≥ s}s
form the (σ,D)-prolongation sequence of a (σ,D)-subvariety of AmV̄a. We define
the m-th arc space of V at a, AmVa, to be this subvariety. We have also that
Φr(AmVa) = AmΦr(V )ψ for all r.r(a)
Proof:
As AmΦr(V̄ ) = Φr(AmV̄ ), we look at AmΦr(V )ψ (a) as an algebraic subvarietyr
of Φr(AmV̄ )ψ (a). We have that Φr+1(V ) ⊂ Φ(Φr(V )) for all r. Since A preservesr
inclusion we have AmΦr+1(V )ψ (a) ⊂ AmΦ(Φr(V )ψ (a)) = Φ(AmΦr(V )r+1 r ψr(a)).
This proves conditions 1. and 2.(a) of 4.4.
Moreover, the maps tr,s : Φr(V ) → Φs(V ) are dominant, and as a is non-
singular, by 4.2, the maps A(tr,s) : AmΦr(V )ψ (a) → AmΦs(V )ψ (a), are dominant.r s
Applying Am to the dominant maps π′1 : Φr+1(V )→ τ(Φr(V )) and π′2 : Φr+1(V )→
τ(Φ σr(V )) , using the hypothesis on a and 4.2, we get
Amπ′1(Am(Φr+1(V )ψ (a))) = Am(π′r+1 1(Φr+1(V ))π′1(ψr+1(a)))
and
Amπ′2(Am(Φr+1(V )ψ (a))) = A ′r+1 m(π2(Φr+1(V ))π′2(ψr+1(a)))
and since π′ (Φ (V ))σ and π′1 r+1 2(Φr+1(V )) have the same Zariski closure, and σ(π
′
1ψr+1(a)) =
π′2ψr+1(a) we get condition 2(b). Hence {Am(tr,s) : AmΦr(V )ψ (a) → AmΦs(V )r ψs(a), r ≥
s} is the (σ,D)-prolongation sequence of a (σ,D)-subvariety W of AmV̄a, where
W (K) = {x ∈ AmV̄a(K) : ψr(x) ∈ AmΦr(V )ψ (a)(K), r ≥ 0} and AmΦr(V )r ψ(a) =
Φr(W ) for all r. We define then AmVa = W .
2
Lemma 4.7. Let U, V be two (σ,D)-subvarieties of an algebraic variety V̄ . Let
a ∈ U(K) ∩ V (K) be a non-singular point of U . Then U = V if and only if
AmΦl(U)ψ (a) = AmΦl(V )ψ for all m, l.l l(a)
Proof:
If AmUa(K) = AmVa(K) for all m, then Φr(AmUa)(K) = Φr(AmVa)(K). Thus,
by 4.6, AmΦr(U)ψ (a)(K) = AmΦr(V )ψ (K). Hence, for all r and m, we haver r(a)
10 RONALD F. BUSTAMANTE MEDINA
AmΦr(U)ψ (a) = AmΦr(V )ψ (a). Lemma 4.3 implies that U and V have the samer r
(σ,D)-prolongation sequence. Hence U = V .
2
Definition 4.8. Let V be a (σ,D)-variety and a a non-singular point of V . We
define the (σ,D)-tangent space Tσ,D(V )a of V at a as follows:
Let Pr be a finite tuple of polynomials generating I(Φr(V )ψ ). Then T (V )r(a) σ,D a
is defined by the equations JPr (ψr(a)) · (ψr(Y )) = 0. In other words, the prolonga-
tion sequence of Tσ,D(V )a is dtl,r : T (Φl(V ))ψ (a) → T (Φr(V ))ψ (a), l ≥ r}, wherel r
T denotes the usual tangent bundle and tl,r the natural projection Φl(V )ψl(a) →
Φr(V )ψ .r(a)
Remark 4.9. Let a be a non-singular point of the (σ,D)-variety V . Then Tσ,D(V )a
is a subgroup of Gna(K), and by the same arguments as above, its prolongation
sequence is (d(tl,r)ψ (a) : T (Φl(V ))ψ (a) → T (Φr(V ))l l ψ ) .r(a) l≥r
Lemma 4.10. Let V be a (σ,D)-variety in Al and a a non-singular point of V .
Then A1Va is isomorphic to T (V )a. Let V̄ be the Zariski closure of V (U) in Al and
m ∈ N; then the map given by lemma 4.1 which identifies the fibers of Am+1V̄a →
AmV̄a with T (V̄ )a restricts to an isomorphism of the fibers of Am+1Va → AmVa
with T (V )a.
Proof:
As remarked before, we can assume that A1V̄ = T (V̄ ). Let b ∈ T (V̄ )a(K).
By definition (a, b) ∈ A1V (U) if and only if ψr(a, b) ∈ T (Φr(V ))(K) for all r. As
T (Φr(V̄ )) = Φr(T (V̄ )), T (Φr(V ))ψ (a) is an algebraic subvariety of Φr(T (V̄ )) andr
ψr(a, b) = (ψr(a), ψr(b)). Hence b ∈ A1Va(K) if and only if b ∈ T (V )b and the first
part of the theorem is proved.
Now we look at the map given in 4.1. In particular, if c ∈ AmVa(K) and r ≥ 0,
by 4.6, ψr(c) ∈ AmΦr(V )ψ (a) and the following diagram commutesr
(Am+1V̄ ) //a c (Am+1Φr(V̄ )ψr(a))ψr(c)
 
T (V̄ ) //a T (Φr(V̄ ))ψr(a)
where the horizontal arrows are ψr and the vertical arrows are the maps given by
4.1 applied to V̄ and Φr(V̄ ). So (Am+1Va)c is identified with Tσ,D(V )a.
2
Notation and Definition 4.11. In [9] the author proved that given a difference-
differential subfield F of K and a ∈ K there is a numerical polynomial Pa/F (X) ∈
Q[X] of degree at most 2, such that for sufficiently large r ∈ N, Pa/F (r) = tr.dg(ψr(a)/F ).
We call the degree of Pa/F the (σ,D)-type of a over F , and the leading coefficient of
Pa/F the typical dimension of a over F , it is denoted dimσ,D(a/F ). For a (σ,D)-
variety V defined over F we define PV = Pa/F where a is a (σ,D)-generic of V
over F . We have that the (σ,D)-type of a over F is 2 if and only if a contains an
element which is (σ,D)-transcendental over F .
Let F = acl(F ) ⊂ U and let a ∈ U and let p = tp(a/F ). We denote by m(p) (or
by m(a/F )) the (σ,D)-type of a over F and we write dimσ,D(p) for dimσ,D(a/F ). If
ZILBER’S DICHOTOMY FOR DIFFERENTIALLY CLOSED FIELDS WITH AN AUTOMORPHIS1M1
p′ is a non-forking extension of p then m(p) = m(p′) and dimσ,D(p) = dimσ,D(p
′).
If A is an arbitrary subset of U we write m(a/A) instead of m(a/acl(A)).
If V is a (σ,D)-variety over K, m(V ) denotes the (σ,D)-type of V . Then, if a
is a (σ,D)-generic of V , m(V ) = m(qftp(a/F )).
Corollary 4.12. . Let V be a (σ,D)-variety in Al, and m ∈ N. Then for a ∈ V (K)
non-singular, the (σ,D)-type of V and AmVa are equal.
Proof:
By 4.6 Φr(AmVa) = AmΦr(V )ψ . But if b is a non-singular point of a varietyr(a)
U , then by 4.10, we have that dim(AmUb) = m dim(U).
2
Remark 4.13. By 4.10, if m = 1 and for a ∈ V (K) non-singular, we have PV =
PT (V ) .a
Lemma 4.14. Let F = acl(F ). Then
(1) m(a, b/F ) = max{m(a/F ),m(b/F )}.
(2) If m(a/F ) = m(b/F ) then dimσ,D(a, b/F ) = dimσ,D(a/F )+dimσ,D(b/Fa).
(3) If m(a/F ) > m(b/F ) then dimσ,D(a, b/F ) = dimσ,D(a/F ).
We borrow the next definitions from [10]
Definition 4.15. (1) Let p be a regular type. We say that p is (σ,D)-type
minimal if for any type q, p 6⊥ q implies m(q) ≥ m(p).
(2) A (σ,D)-variety V is (σ,D)-type minimal if for every proper (σ,D)-subvariety
U , m(V ) < m(U).
Lemma 4.16. Let p be a type and let V be the (σ,D)-locus of p over K If V is
(σ,D)-type minimal then p is regular and (σ,D)-type minimal.
Proof:
Let a be a realization of a forking extension of p to some L = acl(L) ⊃ K.
Let b realize a non-forking extension of p to L. Let U be the (σ,D)-locus of (a, b)
over L. Then the projection on the second coordinate: U → V is dominant, thus
m(a, b/L) ≥ m(V ). Now if a /|̂ Lb, then the (σ,D)-locus of b over acl(La) is a proper
subvariety of V and therefore, by 4.14, m(b/La) < m(V ); from m(a/L) < m(V ),
we deduce m(a, b/L) < m(V ) which is impossible.
2
Lemma 4.17. If p is a type over K, there is a finite sequence of regular types
p1, · · · , pk such that m(p) ≥ m(pi) for all i and p is domination-equivalent to p1 ×
· · · × pk.
Proof:
By 2.12 it suffices to show that given a regular type q, such that p 6⊥ q, there
is a regular type r such that q 6⊥ r and m(r) ≤ m(p). Let a be a realization of
a non-forking extension of p to some L and let b be a realization of a non-forking
extension of q to L such that a /|̂ Lb. Let c = Cb(tp(a/L, b)). Thus c 6∈ acl(L)
and c ∈ acl(Lb). So r = tp(c/L) is non-orthogonal to q and regular (because
c ∈ acl(Lb) and regularity is preserved by algebraicity) . On the other hand, by
12 RONALD F. BUSTAMANTE MEDINA
2.8, there are a1, · · · , al realizations of p such that c ∈ dcl(La1 · · · al). Then, by
4.14, m(r) ≤ m(q).
2
Remark 4.18. By 2.11, in the proof above we can suppose that SU(c/L) = ωi
for i ∈ {0, 1, 2}. Thus, given a type p over K. there is a type q such that it is
(σ,D)-type minimal and has SU -rank ωi.
Lemma 4.19. Let G be a (σ,D)-vector group (that is, a (σ,D)-variety which is a
subgroup of Gka for some k). Then Tσ,D(G)0 is definably isomorphic to G. More-
over, if H is a (σ,D)-subgroup of G, then the restriction of this isomorphism to H
is an isomorphism between H and Tσ,D(H)0.
Proof:
Suppose that G is a (σ,D)-subgroup of Gka. For each r ∈ N, Φr(G) is a subgroup
2 2
of Φr(Gk Gk(r+1) k(r+1)a) = a . Since Φr(G) is an algebraic subgroup of Ga , its defin-
ing ideal is generated by homogeneous linear polynomials, and thus its tangent space
at 0 is defined by the same polynomials. Then the map zr : Φr(Gk ka)→ T (Φr(Ga))
defined by x 7→ (0, x) identifies Φr(Gka) and T (Φr(Gka))0 and it restricts to an iso-
morphism Φr(G) → T (Φr(G))0. Hence (zr : r ≥ 0) identifies the prolongation
sequence of G and the prolongation sequence of Tσ,D(G)0.
For the moreover part, it suffices to note that, by our construction above, the
restriction of zr to Φr(H) is an isomorphism between Φr(H) and T (Φr(H))0.
2
Corollary 4.20. Let G be a (σ,D)-subgroup of Gka. Suppose that for every proper
definable subgroup H of G, m(H) < m(G). Then m(V ) < m(G) for any proper
(σ,D)-subvariety of G. In particular the generic type of G is regular.
Proof:
Let V be a (σ,D)-type minimal (σ,D)-subvariety of G such that m(V ) = m(G).
After possibly replacing V by a translate we may assume that 0 ∈ V and 0 is non-
singular. By Remark 4.13, m(T (V )0) = m(V ) = m(G). Since T (V )0 is a subgroup
of T (G)0 ' G,we obtain T (V )0 = T (G)0. By 4.12, PV = PT (V ) = P0 T (G) = PG.0
Hence V = G. By 4.16, the generic type of G is regular.
2
Lemma 4.21. Let a, c be tuples of U . Let V be the (σ,D)-locus of a over K.
Assume that c = Cb(qftp(a/acl(Kc))). Then there is m ∈ N and a tuple d in
AmVa such that c ∈ K(a, d)σ,D.
Proof:
Let U be the (σ,D)-locus of a over acl(Kc). Then AmUa ⊂ AmVa. As DCFA
eliminates imaginaries every definable set has a canonical parameter. Then c is
interdefinable with the canonical parameter of U which, by 4.7, is interdefinable
over K(a)σ,D with the sequence of the canonical parameters of AmUa over K(a)σ,D.
By quantifier-free stability AmUa is defined with parameters from AmUa.
2
ZILBER’S DICHOTOMY FOR DIFFERENTIALLY CLOSED FIELDS WITH AN AUTOMORPHIS1M3
Lemma 4.22. Let (K,σ,D) be a submodel of (U , σ,D). Let V be a (σ,D)-variety
defined over K and let a ∈ V (U) be a non-singular point. Let b ∈ AmVa. Then
there are b1, b2, · · · , bm = b, such that bi ∈ acl(Ka, b) and each bi is in some
K ∪ {a, bi−1}-definable principal homogeneous space for T (V )a.
Proof:
By 4.1 and 4.10 each fiber ρi+1,i : Ai+1Va → AiVa is a principal homogeneous
space for T (V )a. Then set bi = ρm,i(b).
2
The following lemmas will state the connections between regular non locally
modular types and vector groups.
Lemma 4.23. Let (K,σ,D) be a submodel of (U , σ,D). Let p be a (σ,D)-type
minimal regular type over K such that m(p) = d. If p is not locally modular, then
there are a vector group G and a type q such that:
(1) m(q) = m(G) = d.
(2) (x ∈ G) ∈ q.
(3) p 6⊥ q.
Proof:
By 4.18 we may assume that SU(p) = ωj where j ∈ {0, 1, 2}. By 2.9, enlarging
K if necessary, there are tuples a and c, with a a tuple of realizations of p, tp(c/K)
p-internal, c = Cb(a/acl(Kc)), tp(a/Kc) p-semi-regular and c ∈/ clp(Ka). Let V be
the locus of a over K.
By 4.21, there is a k-tuple d in AmVa(U) such that c ∈ acl(K, a, d). For i =
1, . . . ,m let di = ρm,i(d). Then for each i, di is in some K(adi−1)-definable T (V )
k
a-
principal homogeneous space.
Let m = wp(c/Ka). This means that for any L = acl(L) ⊂ U such that L |̂ Kc,
given a tuple (g1, · · · , gm) realizing p(m) we have that g /|̂i Lc for all i if and only if
g ⊂ clp(Lc).
As c ∈ acl(K, a, d), c 6∈ clp(K, a), tp(c/K) is p-internal; there is j ∈ {1, · · · , k}
such that wp(c/Kadj−1) = m and wp(c/Kadj) ≤ m − 1. Let L = acl(L) ⊂ U
contain Kad − , such that L |̂ C, and (g , · · · , g ) realizing p(m) such that g /|̂j 1 K 1 m i Lc
for all i. Since w (c/Kad − ) > w (c/Kad ), either there is g such that g |̂p j 1 p j k k Ld c,j
or tp(gk/Ldj) forks over L. In both cases, dj and g are dependent over L. Hence
tp(dj/Kadj−1) ⊥6 p.
Let q = tp(dj/Kadj−1). Then we have m(q) = m(H) ≤ m(T (V )a) = m(p),
hence m(p) = m(q).
2
Lemma 4.24. Let p be a regular (σ,D)-type minimal type. If there are a (σ,D)-
vector group G and a type q that satisfy the conclusions of 4.23, then there exists
a (σ,D)-vector group whose generic type is regular, (σ,D)-type minimal and non-
orthogonal to p.
Proof:
We order the triplets ord(G) = {m(G),dimσ,D(G), SU(G)} with the lexicograph-
ical order. We proceed by induction on ord(G).
14 RONALD F. BUSTAMANTE MEDINA
Claim: We may assume that ifH is a proper quantifier-free connected, quantifier-
free definable subgroup of G, then m(H) < m(G).
Proof of the claim: Suppose that m(H) = m(G). Let µ : G → G/H be the
quotient map. By 4.14, ord(G) > ord(G/H). If we replace q by a non-forking
extension of q we may assume that H is defined over the domain A of q. Let a be a
realization of q with tp(a/A) 6⊥ p. As q 6⊥ p, we have either p 6⊥ q0 = qftp(µ(a)/A)
or p 6⊥ q′ = qftp(a/Aµ(a)). If p 6⊥ q0 then m(p) ≤ m(q0) by 4.15, and since
(x ∈ G/H) ∈ q0, m(q0) ≤ m(G/H) ≤ m(G) = m(p). So m(q0) = m(p) and we
apply the induction hypothesis to p, q and G/H. If p 6⊥ q′0 , let b be a realization of
qftp(a/Aµ(a)) such that b |̂ ′′Aµ(a)a. Then a− b ∈ H and p 6⊥ q = qftp(a− b/Ab)
and the same argument applies.
By 4.20 and as q is realized in G and m(p) = m(q) = m(G), q is a generic of G,
and is regular and (σ,D)-type minimal. 2
Corollary 4.25. Let p be regular non locally modular type. Then there is a (σ,D)-
vector group G whose generic type is (σ,D)-type minimal and non-orthogonal to
p.
Proof:
By 4.17 there is a regular type q of minimal (σ,D)-type which is non-orthogonal
to p. By 4.23, q satisfies the hypothesis of 4.24, then there is a (σ,D)-vector group
G whose generic type r is nonorthogonal to q; again by 2.13, then there is such an
r which is non-orthogonal to p.
2
Lemma 4.26. Let G be a (σ,D)-vector group and let p be its generic type. If
p is regular there is a definable subgroup of Ga whose generic type is regular and
non-orthogonal to p.
Proof:
Suppose that G < Gda for some d ∈ N. One of the projections π : G→ Ga must
have an infinite image in Ga. Let a realize p, then π(a) realizes the generic type
of H = π(G); this type is tp(π(a)/K)) which is also regular. Hence H satisfies the
conclusion of the lemma.
2
We have now all we need to prove 3.
Proof of Theorem 3:
By 4.25 there is a (σ,D)-vector group G whose generic type q is regular and
non-orthogonal to p, by 4.26 there is a definable subgroup H of the additive group
whose generic type r is regular and non-orthogonal to q. By transitivity of non-
orthogonality on regular types, p 6⊥ r.
2
5. Partial differential fields with an automorphism
As pointed out in [8], the previous work can be generalised to fields with several
commuting derivations and a commuting automorphism of characteristic zero. We
give the details.
ZILBER’S DICHOTOMY FOR DIFFERENTIALLY CLOSED FIELDS WITH AN AUTOMORPHIS1M5
A differential field is a field equipped with a finite set of commuting derivations
∆ = {D1, ..., Dn} . Then the ring of differential polynomials over K is defined as
the ring of polynomials on the variables Dm11 · · ·Dmnn X, we denote it by K[X]∆
and we can equip it with structure of a differential ring by extending ∆.
Let Ln be the language of differential rings with m derivations, and DFn the
theory of differential fields of characteristic zero with n commuting derivations over
Ln. In [11], McGrail showed that this theory has a model companion, the theory of
differentially closed fields DCFn, and proved that it is a complete ω-stable theory
which eliminates quantifiers and imaginaries. It has a noetherian topology, defined
by zeros of ideals of differential polynomials, known as ∆- topology or Kolchin
topology.
A partial differential field with an automorphism is a differential field with n
commuting derivations and a commuting automorphism. In [8] the author showed
that for the case of characteristic zero, the class of partial differential fields with an
automorphsim has a model companion. We denote it by DnCFA.
The axiomatisation is the following (2.1 of [8]):
Fact 5.1. Let (K,∆, σ) be a differential-difference field. Then (K,∆, σ) is existen-
tially closed if and only if
(1) (K,∆) is a model of DCFn
(2) Suppose V and W are irreducible ∆-closed sets such that W ⊆ V × V σ
and W projects ∆-dominantly onto both V and V σ. If OV and OW are
nonempty ∆-open sets of V and W , respectively, then there is a ∈ OV such
that (a, σa) ∈ OW .
It is not known if this is a first order axiomatisation, but 2.3 of [8] gives us one
that is first order (using characteristic sets of ideals).
As it is proved in [8], the model theory of DnCFA is quite similar to the one of
DCFA: it is not complete but its completions are easily described, they eliminate
imaginaries, are not stable but are supersimple and quantifier-free stable. Section
2 of [4] can be easily generalised to prove that the SU -rank of a generic of a model
of D CFA is ωn+1n .
Now we want to prove a version of 1.1 for DnCFA.
Let U be a saturated model of DnCFA, let E = acl(E) ⊂ U , and let G be a
n
connected ∆-group defined over K = acl(K). Let G(n) = G×Gσ × · · · ×Gσ and
let q : G→ G(n)n such that q(x) = (x, σ(x), · · · , σn(x)).
Let g be a generic of G such that qn(g) is ∆-independent over K. Then qn(g) is
a (∆-)generic of G(n) an thus q (G) is ∆-dense in G(n). This implies that G(n)n is
connected.
Let H be a definable subgroup of G and let H(n) be the ∆-Zariski closure of
q (H) in G(n)n .
Let H̃(n) = {x ∈ G : qn(x) ∈ H(n)}. These groups form a decreasing sequence
of quantifier-free definable subgroups of G containing H. By Noetherianity this
sequence is finite, so there is N such that H̃(N) = ∩ (n)nH̃ . Thus this is the Zariski
closure of H in G.
As in [4], we can prove that [H̃ : H] <∞.
Let Ci be the field of constants of Di . The field of total constants is C = ∩Ci. Let
Fixσ be the fixed field. Since every algebraic subgroup of a vector group is defined
by linear equations, each H̃n is defined by linear differential equations, and thus H
16 RONALD F. BUSTAMANTE MEDINA
is defined by linear (σ,∆)-equations and this implies that it is a (Fixσ ∩ C)-vector
space. Thus, following [4], we can prove the following lemma.
Lemma 5.2.
(1) Let H be a quantifier-free definable subgroup of Gna . Then H is a (Fixσ∩C)-
vector space, so it is divisible and has therefore no proper subgroup of finite
index. This implies that every definable subgroup of Gna is quantifier-free
definable.
(2) Let G be a definable subgroup of Gna , and H a definable subgroup of G.
Then G/H is definably isomorphic to a subgroup of Gla for some l.
We can apply the above lemma to definable subgroups of the additive group,
and following [4] again, obtain the following proposition.
Proposition 5.3. Let G be a definable subgroup of Gna . If G has infinite dimension
then SU(G) ≥ ω.
In [8], the author proved the dichotomy for finite dimensional types.
Fact 5.4. Let (U , σ,D) be a saturated model of DnCFA and let K = acl(K) ⊂ U .
Let p ∈ S(K) be a finite dimensional type of SU -rank 1. Then p is either 1-based
or non orthogonal to Fixσ ∩ C
Now in order to define arc spaces for a (σ,∆)-variety we must define the differ-
ential prolongation of an algebraic variety. We use in this case the approach of [10]
(section 2).
Let (K,∆ = (D1, · · · , Dn)) be a differential field. DefineKm = K[η , · · · , η ]/(η , · · · , η )m+11 n 1 n .
Let E : K → Km defined by:
∑ 1
a→7 Dα1 · · ·Dαn(a)ηα11 n 1 · · · ηαnα ! · · ·α ! n
0≤ 1 nα1+···+αn
If V is an algebraic variety defined over K, the m-th ∆-prolongation τmV of V
is the Weil restriction of V ⊗E Km from Spec(Km) to Spec(K).
The reduction maps Kl → Km for l ≥ m imply that the prolongations form a
projective system πl,m : τl → τm. If we identify τ0 with the identity and denote
πm, 0 as πm, we obtain the projection πm : τmV → V . The map ∇m : V → τmV
defined by ∑ 1
x→7 Dα11 · · ·Dαn
α1 αn
α ! · · ·α ! n
(x)η1 · · · ηn
≤ ··· 1 n0 α1+ +αn
is a section of πm and ∇m(V ) is ∆-dense in τmV .
Using the definition of qm and Sm(V ) from the previous section we define Φ(V ) =
τm(Sm(V ))(= Sm(τmV )) and ϕ = ∇m ◦ qm.
Now we can define arc spaces of (σ,∆)-varieties as before:
We extend σ and Di to K
(m) by defining σ(ηj) = ηj and Diηj = 0. Since σ
and Di commute, we can identify Ar(Sm(V ))(K) with Sm(Ar(V ))(K) and we can
assume that Ar(Φm(V ))(K) = Φm(Am(V ))(K).
If X is a (σ,∆)-variety given as a (σ,∆)-closed subset of an algebraic variety X̄,
we define Φm(X) as the Zariski closure of ψm(X) in Φm(X̄). X is determined by
the prolongation sequence {tl,m : Φl(X)→ Φm(X)) : l ≥ m}.
ZILBER’S DICHOTOMY FOR DIFFERENTIALLY CLOSED FIELDS WITH AN AUTOMORPHIS1M7
We extend the previous notion of non-singular point, and as before we can show
that if V be a (σ,D)-variety given as a closed subvariety of an algebraic variety V̄ .
If m ∈ N and a ∈ V (K) is a non-singular point. Then {Am(tr,s) : AmΦr(V )ψr(a) →
AmΦs(V )ψ (a), r ≥ s} form the (σ,D)-prolongation sequence of a (σ,D)-subvarietys
of AmV̄a.
So we define the m-th arc space of V at a, AmVa, to be this subvariety.
Now we remark that in [9], the author proved his theorem for a field with n
derivations and k automorphisms, so for our case subfield F of K and a ∈ K there
is a numerical polynomial Pa/F (X) ∈ Q[X] of degree at most n + 1, such that for
sufficiently large r ∈ N, Pa/F (r) = tr.dg(ψr(a)/F ). This implies that the notion of
(σ,∆)-type if a (σ,∆) -variety is well defined and it is less or equal than n+ 1.
If p is a type over K, there is a finite sequence of regular types p1, · · · , pk such
that m(p) ≥ m(pi) for all i and p is domination-equivalent to p1 × · · · × pk. As in
the previous section, we can prove that given a type p over K and a regular type q,
such that p ⊥6 q, there are c and L such that the type r = tp(c/L) is regular, q 6⊥ r,
m(r) ≤ m(p) and SU(r) = ωi for i ∈ {0, 1, · · · , n+ 1}. This is essentially what we
need to prove 4.23.
The other propositions can be translated almost verbatim to the case of several
derivatives, thus obtaining the following version of 4.26.
Lemma 5.5. Let G be a (σ,∆)-vector group and let p be its generic type. If p
is regular there is a definable subgroup of Ga whose generic type is regular and
non-orthogonal to p.
Putting 5.3, 5.4 and 5.5 together we obtain Zilber’s dichotomy.
Theorem 4. Let (U , σ,∆) be a saturated model of DnCFA and C its field of total
constants. Let K = acl(K) ⊂ U and let p be a type over K. If p is a type of
SU -rank 1, then it is either 1-based or non orthogonal to Fixσ ∩ C.
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Email address: ronald.bustamante@ucr.ac.cr
Escuela de Matemáticas, Universidad de Costa Rica, Sede Rodrigo Facio, 2060, San
José, Costa Rica