Topological large fields, their generic expansions todifferential fields and transfer results.

Pablo Cubides (Universite de Caen)Francoise Point (FNRS-FRS, UMons)

Konstanz, October 3rd 2018.

Outline of the talk

dp-minimal fields with a generic derivation,

Transfer results : elimination of imaginaries, continuous defin-able functions, open core,

Applications to dense pairs,

Further directions.

dp-minimal fields

Definition

A theory T is not dp-minimal if there is a model M of T , aij ∈ Mand uniformly unary definable sets Xi , Yj ⊆ M, i , j ∈ N, such thataij ∈ Xi ′ ↔ i = i ′,aij ∈ Yj ′ ↔ j = j .′

A structure is dp-minimal if its theory is.

Examples of dp-minimal fields

Let K := (K ,+, ·,−, 0, 1).• (K, <) an ordered real-closed field, ie a model of RCF [o-minimaltheory]• (K, v) a non-trivially valued algebraically closed field, respectivelyof ACVF [C -minimal theory],• (K, v) a p-adically closed valued field of rank d , respectively ofpCFd [p-minimal theory]• (K, <, v) an ordered valued real-closed field, respectively of RCVF[weakly o-minimal theory].

and more...

dp-minimal fields

Theorem (Johnson)

If K := (K ,+, ·, 0, 1, · · · ) is an expansion of an infinite field with adp-minimal theory but not strongly minimal, then K can be endowedwith a non-discrete Hausdorff definable field topology, namely K hasa uniformly definable basis of neighbourhoods of zero compatiblewith the field operations, etc · · ·

Moreover, Johnson shows that any definable subset of K has finiteboundary and every infinite definable set has non-empty interior (soK eliminates ∃∞).Furthermore, the topology on K is induced either by a non-trivialvaluation or an absolute value.

Correspondences

Now for a dp-minimal field K, we will describe a generalisation ofa cell decomposition theorem due to L. Mathews (for certain topo-logical fields).

Definition

Let E , F be two definable subsets of Kn, then a correspondence fis a definable subset graph(f ) of E × F such that

0 < |{y ∈ F : (x , y) ∈ graph(f )}| <∞, forall x ∈ E .

A correspondence f is an m-correspondence if for all x ∈ E , |{y ∈F : (x , y) ∈ graph(f )}| = m.

dp-minimal fields-definable sets

Let X be a A-definable subset of Kn with A a subset of K , then:

Theorem (Simon-Walsberg)

There a finitely many A-definable subsets Xi with X =⋃Xi such

that Xi is the graph of a A-definable continuous m-correspondencef : Ui ⇒ Kn−d , where Ui is a A-definable open subset of Kd , forsome 0 ≤ d ≤ n.

Conventions: if d = 0, f : K 0 ⇒ Kn−d , then graph(f ) is identifiedwith a finite set andif d = n, f : U ⇒ K 0, graph(f ) is identified with U (an open subsetof Kn).

Note that when acl = dcl, we may replace “correspondence” by thegraph of a definable function.

dp-minimal fields-dimension

Let K be a dp-minimal field and let X be a A-definable subset ofKn. We have several notions of dimensions:

• the topological dimension:let X ⊆ Kn, then dim(X ) := max{` : there is a projection π :Kn → K ` such that π(X ) has non-empty interior}.

• the acl-dimension (acl−dim), defined as follows: acl−dim(u/A) :=min{` : there is a subtuple d of u of length ` such that u ∈acl(A, d)}. Then acl−dim(X/A) := max{acl−dim(u/A) : u ∈ X}.Note that it is not assumed that acl has the exchange.

dp-minimal fields-dimension

Theorem ( Simon-Walsberg )

Then dim(X ) = acl− dim(X )(= dp − rank(X )).

Let fr(X):=X \ X , where X denotes the closure of X .

Theorem (Simon-Walsberg)

dim(fr(X )) < dim(X ).

dp-minimal fields with a generic derivation

From now on, K := (K ,+,−, ·, 0, 1, · · · ) denote a dp-minimal fieldof characteristic 0 and assume that K is not strongly minimal. Fur-thermore, we will assume that:

• the language L is a relational expansion of the ring (field) languageand every relation and its complement is the union of an algebraicset and an open subset.

• The theory T of K admits quantifier elimination in the languageL.

Examples

Let L be the language of fields. Let div be a binary relation.

1 Let L< := L ∪ {<}, then RCF admits quantifier elimination(Tarski),

2 Let Ldiv := L∪{div}, then ACVF admits quantifier-elimination(Robinson).

3 Let L<,div := L< ∪ {div}, then RCVF admits quantifier-elimination (Cherlin-Dickmann).

4 Let Lp := L∪ {div , c1, · · · , cd ,Pn; n ≥ 1}, then pCFd admitsquantifier elimination in Lp (Macintyre, Prestel-Roquette).

In all the above cases, the relations and their complements satisfythe hypothesis to be the union of an open set with an algebraic set.

dp-minimal fields with a generic derivation-some notation:

We consider the generic expansion of K with a derivation δ, namelywe put no a priori continuity assumptions on δ. Denote by Lδ :=L ∪ {δ} and Tδ the Lδ-theory T∪ {δ is a derivation }.

For a ∈ K and m > 0, we let

δm(a) denote the mth-derivative of a, m ≥ 1, with δ0(a) = a,

Jetm(a) = δm(a) = (δ0(a), δ1(a), δ2(a), . . . , δm(a)), and

for X ⊂ K , Jetm(X ) = {δm(a) : a ∈ X}.

dp-minimal fields with a generic derivation-some notation:

By assumption on L, any Lδ-term t(x) with x = (x1, . . . , xn), isequivalent, modulo the theory of differential fields, to an L-termt∗(δm1(x1), · · · , δmn(xn)) for some (m1, · · · ,mn) ∈ Nn.

So with any Lδ-quantifier-free formula ϕ(x), we may associate anequivalent Lδ-formula ϕ∗(δm(x)), m ∈ N, (modulo the theory of dif-ferential fields) where ϕ∗ is a L-quantifier-free formula which arisesby uniformly replacing every occurrence of δm(xi ) by a new vari-able ymi in ϕ with the following choice for the order of variablesϕ∗(y0

1 , · · · , ym1 , · · · , y0n , · · · , ymn ). So we get

ϕ(x1, . . . , xn)⇔ ϕ∗(δm(x1), . . . , δm(xn)).

Scheme (DL)

Let T as before, K |= T and χ(x , y) be an L-formula such that forany a ⊂ K , χ(K , a) is an open neighbourhood of 0 in K .Set T ∗δ := Tδ ∪ (DL), where (DL) is the following list of axioms:

Let a := (a1, · · · , an) ai ⊂ K , 1 ≤ i ≤ n, n ≥ 1, set

Wa := χ(K , a1)× · · · × χ(K , an).

K satisfies (DL) if for every n ≥ 1, for every differential polynomialf (X ) ∈ K{X}, with f (X ) = f ∗(X , δ(X ), . . . , δn(X )) and for anya ⊂ K , we have:

∃α(

(f ∗(α) = 0∧s∗f (α) 6= 0)⇒(∃z(f (z) = 0∧sf (z) 6= 0∧(δ(z)− α) ∈Wa

)).

Axiomatisation of differential t-large e.c. topological fieldsof characteristic 0

Under the further hypothesis, called t-large–it adapts in this topo-logical setting the property of largeness (Pop)-, the theory T ∗δ isconsistent and axiomatizes the class of existentially closed modelsof Tδ. In this particular setting, we get:

Theorem (Guzy-P)

Let T be a dp-minimal theory of t-large L-fields of characteristic 0,admitting quantifier elimination.Then T ∗δ is the model-completion of Tδ and admits quantifier elim-ination.

The above theorem was stated for a larger class of topological fields.

Examples

• We obtain for the theory T ∗δ :

1 CODF = RCF∗δ in case T = RCF,

2 RCVF∗δ in case T = RCVF (an expansion of CODF),

3 pCF∗δ in case T = pCFd ,

4 ACVF0,0∗δ in case T = ACVF0,0 (an expansion of DCF0),

First properties (direct consequences of the axiomatisation)

Using the fact that T ∗δ admits q.e. (and the forgetful functor), onecan observe:

• (Guzy-P.) If T is NIP, then T ∗δ is NIP.

• (Chernikov, 2015) If T is distal, then T ∗δ is distal.

Let K |= T ∗δ and denote by CK its subfield of constants.Using the axiomatisation (respectively the geometrical axiomatisa-tion), two observations:

• Then CK is dense in K .

• (Brouette, Cousins, Pillay, P.–in case L is the language of rings–)Then CK |= T .So we get an elementary pair (K ,CK ) of models of T .

Order of a definable set

Since T ∗δ admits quantifier elimination, every Lδ-definable set X ⊆Kn is of the form Jet−1

m (Y ) for some quantifier-free L-definable setY ⊆ K (m+1)n.

DEFINITION (Order)

Let X ⊆ Kn be an Lδ-definable set. The order of X , denoted byo(X ), is the smallest integer m such that X = Jet−1

m (Y ) for someL-definable set Y ⊆ K (m+1)n.

Open core

Property (?): For any X ⊆ Kn Lδ-definable non-empty subset, thereis an integer m ≥ o(X ) and an L-definable set Z ⊆ K (m+1)n suchthat

1 x ∈ X if and only if Jetm(x) ∈ Z and

2 Z = Jetm(X ).

Note that equivalently in Property (?) one can require that m =o(X ).

Open core-continued

Lemma (C-P)

Property (?) is equivalent to: T ∗δ has L-open core.

(⇒) one shows that given an Lδ-definable set X , its closure X isL-definable.Claim: X = π(Z ), where Z has the property (?) and π is theprojection sending each block of (m + 1) coordinates to its firstcoordinate.

Open core-continued

(⇐) Conversely, if the theory T ∗δ has L-open core, then:

take Y ⊂ K (o(X )+1)n be an L-definable set such that X = Jet−1o(x)(Y ).

Set Z := Y ∩ Jeto(X )(X ). Since Jeto(X )(X ) is both closed and Lδ-definable, it is L-definable (T ∗δ has open core).So the set Z is L-definable. Since

Jeto(X )(X ) ⊆ Z ⊆ Jeto(X )(X ),

both properties (1) and (2) are easily shown.

Elimination of imaginaries

Let G be a collection of sorts of Leq. We let LG denote the restrictionof Leq to the home sort together with the new sorts in G and theirrespective quotient maps.

Theorem (C-P)

Suppose that T admits elimination of imaginaries in LG . If T ∗δ hasL-open core, then T ∗δ admits elimination of imaginaries in LGδ .

We follow an argument of Marcus Tressl to show EI on CODF.

Elimination of imaginaries

Proof:Fix a model K of T ∗δ and let X ⊆ Kn be a non-empty Lδ-definableset.We will show that X has an Lδ-code in G(K ), namely, there is atuple e ∈ G(K ) such that for all σ ∈ AutLδ(K )

σ(X ) = X if and only if σ(e) = e.

Observation 1: every L-definable set has an Lδ-code in G(K ).

X := Jet−1o(X )(Jeto(X )(X )).

By the open core assumption, Jeto(X )(X ) is L-definable.

We proceed by induction on dim(Jeto(X )(X )).

Elimination of imaginaries

Conditional to having L-open core, we obtain the following corollar-ies:

For T = RCF, yet another proof that CODF admits elimina-tion of imaginaries in the language of differential fields.

For T = ACVF, T = RCVF and T = pCF, a proof thatT ∗δ has elimination of imaginaries in the language LGδ whereG corresponds to the so called geometric sorts (by the corre-sponding results of Haskell-Hrushovski-Macpherson, Mellor andHrushovski-Martin-Rideau respectively).

Continuous definable functions and open core

Theorem (C-P)

Assume that aclL = dclL (finite Skolem functions) for models of T .Let X ⊆ Kn be an L-definable set and f : X → K be a continuousLδ-definable function. Then f is L-definable.

COROLLARY

CODF has L-open core.

Continuous definable functions and open core

Let LRCVF and LpCFdbe the languages in which RCVF and pCFd

eliminate quantifiers respectively. Let LΓ be the 2-sorted language

(K ,L){(Γ ∪ {∞},Loag ) if L := LRCVF

(Γ ∪ {∞},LPres) if L := LpCFd

v : K → Γ ∪ {∞}.

Let LΓ,δ be the extension of LΓ in which we replace L by Lδ in thevalued field.

Theorem (C-P)

Let T be RCVF or pCFd . Let K be a model of T ∗δ . Then the LΓ,δ

theory of K has quantifier elimination.

Continuous definable functions and open core

Corollary (C-P)

Let T be RCVF or pCFd . Let K be a model of T ∗δ . Then everyLΓ,δ-definable subset X ⊆ Γ ∪ {∞} is LΓ-definable.

Theorem (C-P)

Assume that aclL = dclL (finite Skolem functions) for models ofT . Let X ⊆ Kn be an L-definable set and f : X → Γ ∪ {∞} be acontinuous LΓ,δ-definable function. Then f is LΓ-definable.

Corollary (C-P)

RCVF∗δ and pCF∗δ have L-open core.

Applications to dense pairs

Let L2 := L ∪ {P} where P is a new unary predicate P.Let T 2 be the L2-theory of dense elementary pairs (K ,F ).

Recall that if K is a model of T ∗δ , then (K ,CK ) is a model of T 2.

Assume from now that T is geometric.

Theorem (van den Dries/Berenstein-Vassiliev/Fornasiero/....)

The theory T 2 is complete.

COROLLARY (C-P)

Every model (K ,F ) of T 2 has an L2-elementary extension (K ∗,F ∗)such that K ∗ is a model of T ∗δ with constant field CK∗ = F ∗.

Applications to dense pairs

COROLLARY (C-P)

Assume that Tδ has open core. Then T 2 has L-open core.

COROLLARY (Boxall-Hieronymi/Fornasiero)

If T is RCF, RCVF or pCF, then T 2 has L-open core.

Applications to dense pairs

Theorem (Hieronymi, Nell)

Let T be an o-minimal theory extending the theory of orderedabelian groups. Then the theory T 2 is not distal.

Question: (Simon) Does T 2 admit a distal expansion?

Theorem (Nell)

Let F be an ordered field, A be an ordered F -vector space and Ba dense subspace. Let L denote the language of ordered F -vectorspaces. Then, the L2-theory of (A,B) has a distal expansion.

Applications to dense pairs

Theorem (Chernikov)

Assume that T is distal. Then T ∗δ is distal.

Let T 2δ be the L2

δ-theory extending T ∗δ by the axiom

∀x(P(x)↔ δ(x) = 0),

i.e., P is interpreted as the constant field.

Corollary (C-P)

The theory T 2δ is a distal expansion of T 2.

In particular, the theories of dense pairs of real-closed fields, densepairs of p-adically closed fields and dense pairs of real closed valuedfields admit a distal expansion.

Further directions

Let K be a model of T ∗δ .

Show that if f : X ⇒ K is a continuous Lδ-definable correspon-dence and X ⊆ Kn is L-definable, then f is L-definable.

Develop the formalism of T ∗δ when T is a theory in a multi-sorted language. This might provide a way to deal with valuedfields such as C((X )) and R((X )).

Extend this formalism to some dp-minimal expansions of fields.

Thank you for your attention.