geometric axioms for existentially closed hasse fields

Annals of Pure and Applied Logic 135 (2005) 286–302

www.elsevier.com/locate/apal

Geometric axioms for existentially closedHasse fields

Piotr Kowalski∗Wrocław University, Poland

University of Illinois at Urbana-Champaign, United States

Received 11 September 2004; received in revised form 22 February 2005; accepted 23 February 2005Available online 17 May 2005

Communicated by A.J. Wilkie

Abstract

We give geometric axioms for existentially closed Hasse fields. We prove a quantifier eliminationresult for existentially closedn-truncated Hasse fields and characterize them as reducts ofexistentially closed Hasse fields.© 2005 Elsevier B.V. All rights reserved.

Keywords:Hasse derivation; Model companion; Prolongation

0. Introduction

The purpose of this paper is to give geometric axioms for SCHp,e (the model companionof the theory of fields of positive characteristicp with e commuting iterative Hassederivations) in the spirit of the axiomatizations of ACFA [2], DCF0 [9] or DCFp,n [5].Axioms for SCHp,e were given in [15] and they are actually fairly simple, i.e. a field asabove is existentially closed if and only if it is strict and separably closed of inseparabilitydegreee; see [15, 1.1].

Supported by an NSF FRG postdoc and Polish KBN grant 2P03A 018 24.∗ Corresponding address: Instytut Matematyczny, University of Wrocław, pl. Grunwaldzki 2/4, 50–384Wrocław, Poland.

E-mail address:[email protected].

0168-0072/$ - see front matter © 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.apal.2005.02.003

http://www.elsevier.com/locate/apal

P. Kowalski / Annals of Pure and Applied Logic 135 (2005) 286–302 287

One could wonder why we need geometric axioms—possibly (and actually) morecomplicated than Ziegler’s ones. However, Ziegler’s axioms do not say much about thesolvability of Hasse differential equations. Specifically, we know that in models of SCHp,e

all solvable (in an extension) systems of Hasse differential equations and inequalities canbe solved. But we still do not know which systems are solvable. As was pointed out to meby Margit Messmer and Anand Pillay, even very simple Hasse differential equations mayhave no generic solutions, i.e. the equationX′ = X has 0 as its unique solution, so thesystemX′ = X ∧ X = 0 is not solvable.

Therefore the situation is different than in the differential case. From Blum’s axioms [1]for differentially closed fields of characteristic 0 we know that each non-explicitlyinconsistent differential equation has a generic solution in a differentially closed fieldof characteristic 0. In the case of differentially closed fields of positive characteristic,Wood’s axioms [13, 2.5] tell us that each differential equation satisfying a certain separantcondition (needed to get rid of inseparability issues) has a generic solution. The same istrue in the case of derivations of Frobenius [5].

Our axioms for SCHp,e give certain geometric criteria for deciding whether a givensystem of Hasse differential equations and inequalities has a solution. We also work withn-truncated Hasse derivations (as in [10]) and prove that their theory has a model compan-ion, SCHp,e,n, which is a theory intermediate between the theory of separably closed fieldsof inseparability degreee and SCHp,e. Therefore SCHp,e,n is stable and not superstable.We also show that SCHp,e,n does not have quantifier elimination in the Hasse field lan-guage, but eliminates quantifiers after adding thep-th root function as in the case of DCFp.

1. Basic notions about Hasse derivations

We do not use any symbol for multiplication or composition. It is always clear fromthe context which operation is taken. For examplef (a)n denotes then-th power of f (a)

and f n(a) denotes then-th composition of f with itself applied toa. To avoid possibleconfusion with the set of then-th powers, then-th Cartesian product of a setA is denotedby A×n.

All the rings considered in this paper are commutative, with unity and of primecharacteristicp > 0. Let e be a fixed positive integer. In this section we recall somedefinitions, mainly from [15]. We use the termHasse fieldinstead ofD-field.

A sequenceD = (Di : R → R)i0 is a Hasse derivation on a ringR, if D0 = id andthe corresponding map

R a →∞∑

i=0

Di (a)Xi ∈ R[[X]]

is a ring homomorphism.D is iterative, if for any i , j

Di D j =(

i + j

i

)Di+ j .

288 P. Kowalski / Annals of Pure and Applied Logic 135 (2005) 286–302

It is clear that it is enough to check the above equality on a set of generators ofR. For therest of the paper we will denote the binomial coefficient

(i+ ji

)by ci, j . Note that

ci, j ≡ 0(mod p)

for i , j < pn such thati + j pn (n ∈ N). We will regardci, j as an element ofFp (thefield of p elements).

For two sequencesi = (i1, . . . , i e), j = ( j1, . . . , je) of natural numbers, we define

ci,j := ci1, j1 . . . cie, je.

For eachn we have also anR-linear map:

cn : R×pn → R×p2n, cn(a0, . . . , apn−1) = (ci, j ai+ j )0i, j <pn .

Note thatci, j ai+ j always makes sense, since fori , j < pn andi + j pn, ci, j = 0, sowe putci, j ai+ j = 0 in that case.

For example, ifn = 1, p = 3, then we get a map of the form

R×3 (a0, a1, a2) →a0 a1 a2

a1 2a2 0a2 0 0

∈ R×9.

For anym, let us denote(D0, . . . , Dm−1) by D<m. In the following, we also considerD<m

as a map fromR to R×m andD as a map fromR to R×ω.Let D1, D2 be Hasse derivations onR. We say thatD1, D2 commute if for anyi , j ,

D1,i D2, j = D2, j D1,i .

It is again clear that the above equality needs to be checked on a set of generators ofRonly. A Hasse ringis a ring with a finite number of commuting iterative Hasse derivations.For a Hasse ring(R, D1, . . . , De), thering of constants of Ris the ring of common zeroesof D1,1, . . . , De,1 and thering of absolute constants of Ris the ring of common zeroesof D1, . . . , De. It is always true thatRp is a subring of constants andRp∞

is a subringof absolute constants.(R, D1, . . . , De) is strict if the constants ofR coincide with Rp.It follows from equation (2.2) in the proof of [15, 2.4] that if (K , D1, . . . , De) is a strictHasse field, then for eachn, common zeroes ofD1,<pn, . . . , De,<pn coincide withK pn

.For the notions ofp-independence,p-basis and the inseparability degree of a field of

characteristicp, see [3]. If (K , D1, . . . , De) is a Hasse field, then ap-basisb1, . . . , be ofK is canonicalif

Di,m(bj ) =

1 if m = 1 andi = j0 otherwise.

Following [16], we say that ap-basis iscanonical of depth kif the above holds form < pk.It is shown in [16] that every strict Hasse field of degree of imperfectione has a canonicalp-basis of depthk for any natural numberk. Hence [16] every ω-saturated strict Hassefield of degree of imperfectione has a canonicalp-basis.

We will also need truncated Hasse derivations (as in [10]).


A finite sequence of maps(Di : R → R)i<pn is called ann-truncated Hasse derivation(n 1 always) on a ringR if D0 = id and the corresponding map

R a →pn−1∑i=0

Di (a)Xi ∈ R[X]/(X pn)

is a ring homomorphism.We get a natural notion of ann-truncatediterativeHasse derivation:

Di D j =

ci, j Di+ j for i + j < pn

0 for i + j pn,

since againci, j = 0 if i , j < pn andi + j pn.For example an iterative 1-truncated Hasse derivation is given by a sequence

(id, D, D2/2, . . . , Dp−1/(p − 1)!)for a derivationD such thatDp = 0.

We define ann-truncated Hasse ringin an obvious way. Obviously, the notion of acanonicalp-base (of depthk) makes sense for ann-truncated Hasse field (ifk n). Alsoconstants and absolute constants are defined in the same way as in the Hasse case (e.g. fora 1-truncated Hasse ring, constants coincide with absolute constants).

Let i ∈ N×e (we considerN×e as a monoid, i.e. addition makes sense). For a Hasse ring(R, D1, . . . , De), we define

Di := D1,i1 . . . De,ie.

From now on we denote a cartesian power of any mapf by the same symbolf . Inparticular it applies to the mapscn, D<pn in the next lemma.

Lemma 1.1. Let (R, D1, . . . , De) be a Hasse ring (e= 1 in (ii) and(iii) ).

(i) For i, j ∈ N×e we have

Di Dj = ci,j Di+j.

(ii) If b ∈ R×m, then

D<pnD<pn(b) = cnD<pn(b).

(iii) Assume that R= K is a field. LetD′<pn be an n-truncated Hasse derivation on a

field L such that(K , D<pn) ⊂ (L, D′<pn). If there exists a tuple a⊂ L such that

L = K (D′<pn(a)) andD′

<pn is iterative on a, thenD′<pn is iterative.

Proof. (i)

Di Dj = D1,i1 D1, j1 . . . De,ie De, je = ci1, j1 . . . cie, je D1,i1+ j1 . . . De,ie+ je = ci,j Di+j.

(ii) It is a straightforward generalization of the iterativity condition.(iii) Sincecn is defined by a matrix overFp, D′

<pn andcn commute. Therefore (by (ii))

D′<pnD′

<pnD′<pn(a) = D′

<pncnD′<pn(a) = cnD′

<pnD′<pn(a).


HenceD′<pn is iterative on the setK ∪ D′

<pn(a); therefore it is iterative onK [D′<pn(a)].

By formula (2.1) in the proof of [7, 2.3],D′<pn is iterative onL.

For a Hasse ring(R, D1, . . . , De) and a tuple of variablesX, there exists a Hasse ringof Hasse polynomialsRX (see Section 3 of [7]). As a ring,

RX := R[Xi | i ∈ N×e] (let X = X(0,...,0)),

where

Di(Xj) = ci,jXi+j.

If e = 1, thenRX (as a ring) is justR[X, X′, . . . , X(n), . . .].It is easy to see that for eachn, the ring

RXn := R[Xi | i ∈ 0, 1, . . . , pn − 1×e]is ann-truncated Hasse ring withD1,<pn, . . . , De,<pn .

The notion of an (n-truncated)Hasse idealis the obvious one and one can see thatquotients of (n-truncated) Hasse rings by (n-truncated) Hasse ideals have a natural structureof (n-truncated) Hasse rings.

2. Truncated Hasse fields

In this section we deal with truncated Hasse fields. The main purpose is to provide a stepin the proof of the main theorem (4.3). We also characterize existentially closed truncatedHasse fields as reducts of existentially closed Hasse fields.

We will need a fact about expansions ofn-truncated iterative Hasse derivations to Hassederivations (2.3). First we need a version of Lemma 4 from [16].

Lemma 2.1. Let (K , D1, pn, . . . , De, pn, De+1,<pn , . . . , De′,<pn) satisfy its part of theaxioms for strict truncated Hasse fields (n 1, e e′). Let a ∈ K be such that for allm < n

Di,pn D j ,pm(a) = 0

for i e, j e′. Then there is a′ ∈ K such thatDj ,pn(a′) = 0 for all j eDj ,<pn(a′) = Dj ,<pn(a) for all j e′.

Actually Lemma 4 in [16] is formulated for strict Hasse fields but the proof also works inthe truncated case.

The next lemma slightly generalizes [16, 2]; however, its proof follows from the proofof [15, 2.1].

Lemma 2.2. Let D1, . . . , De be commuting derivations on a field K such that Dp1 =

. . . Dpe = 0 and let C denote the constants of(K , D1, . . . , De). Then the following are

equivalent:

(i) There exist a1, . . . , ae ∈ K such that Di (aj ) = δij .


(ii) [K : C] = pe.

Moreover, each sequence as in(i) generates K over C.

Proof. By [16, 2], we need only prove that (i) implies (ii).For i e let Bi denote the field of constants of(K , D1, . . . , Di ) (so Be = C). For

i < e, we haveDi+1(Bi ) ⊂ Bi andai+1 ∈ Bi \ Bi+1. Since[K : C] pe [15, 2.1], andap

i+1 ∈ Bi , the result follows. Abusing the language a little bit, we will say that for a field extensionK ⊂ L a subset

B ⊂ L is a p-basis of L over Kif

bα11 . . . bαn

n | αi < p, i n < ω, b1, . . . , bn ∈ Bis a basis ofL overK L p (as a linear space).

Theorem 2.3. Assume that(K , D1, . . . , De) is a Hasse field which has a canonicalp-basis. Suppose that L is a finitely generated extension of K and there are commutingiterative n-truncated Hasse derivationsD′

1,<pn, . . . , D′e,<pn on L such that

(K , D1,<pn, . . . , De,<pn) ⊂ (L, D′1,<pn , . . . , D′

e,<pn).

Then there is(D1, . . . , De), a Hasse field structure on L such that

(K , D1, . . . , De), (L, D′1,<pn , . . . , D′

e,<pn) ⊂ (L, D1, . . . , De),

i.e. (L, D1, . . . , De) is an extension of(K , D1, . . . , De) and an expansion of(L, D′

1,<pn, . . . , D′e,<pn).

Proof. By definition of the canonicalp-basis,K has inseparability degreee. By 2.2,(K , D1, . . . , De) is strict.

We will use [15, 2.1] several times. It is stated for extensions of Hasse fields; howeverwhat is used in the proof is only thatD1,1, . . . , De,1 is a sequence of commuting derivationssuch that each of them composed with itselfp times vanishes.

Let C denote the constants ofL and B be a canonicalp-basis ofK . By [15, 2.1],[L : C] pe andC is linearly disjoint fromK over K p, so [L : C] = pe. ThereforeBis also ap-basis ofL overC. Let B0 be a p-basis ofC over L p. ThenB1 := B ∪ B0 isa p-basis ofL. It is finite, sinceL is finitely generated overK . By [15, 4.1], there exists(D′′

1, . . . , D′′e′), a Hasse structure onL havingB1 as its canonicalp-basis which implies

that

(K , D1, . . . , De), (L, D′1,<p, . . . , D′

e,<p) ⊂ (L, D′′1, . . . , D′′

e).

Now we want to find a newp-basis onL which will be a canonicalp-basis of depth 2 for

L := (L, D′1, p, . . . , D′

e, p, D′′e+1,<p, . . . , D′′

e′,<p).

Let a ∈ B0. It satisfies the assumption of2.1for L. Let B00 = a′ : a ∈ B, where eacha′is as in the conclusion of2.1. ThenB2 := B ∪ B00 is a canonicalp-basis of depth 2 forL.We use this basis to get(D′′′

1, . . . , D′′′e′), a Hasse structure onL such that

(K , D1, . . . , De), (L, D′1,<p2, . . . , D′

e,<p2) ⊂ (L, D′′′1, . . . , D′′′

e).


We continue doing the last step of the proof until we getD(n+1)1 , . . . , D(n+1)

e′ .

(D1, . . . , De) := (D(n+1)1 , . . . , D(n+1)

e ) works.

Recall that every strict Hasse field (e Hasse derivations) of degree of imperfectione hasa canonicalp-basis of depthk for any natural numberk [16]. Actually, the proof from [16]also gives a version in the truncated case:

Theorem 2.4. Let (K , D1,<pn, . . . , De,<pn) be a strict n-truncated Hasse field. If e isthe inseparability degree of K , then(K , D1,<pn , . . . , De,<pn) has a canonical p-basis ofdepth n.

We are interested in extensions ofn-truncated Hasse fields.

Fact 2.5. Any n-truncated Hasse field has a strict extension.

Proof. Let (K , D1,<pn , . . . , De,<pn) be ann-truncated Hasse field andC denote its fieldof constants. Obviously,K p ⊂ C. By the usual chain procedure, it is enough to prove thefollowing claim.

Claim. If there is an a∈ C \ K p, then there exists an n-truncated Hasse field extensionK ⊂ L such that a∈ L p and Lp ⊂ K .

Proof of Claim. In this proof we exercise the general tactic of this paper; i.e. first we provethe statement in the case of one (truncated) Hasse derivation, then we point out how theproof extends to the general case.

We work with the ringK Xn of n-truncated Hasse polynomials (in one variable). Inthis caseK Xn = K [X, X′, . . . , X(pn−1)]. Let I be the ideal ofK Xn generated by theset

X(pn−1+k), (X(i ) − Dpi (a)1/p)pi | 0 k < (p − 1)pn−1, 0 i < pn−1,where

pi =

1 if Dpi (a) ∈ K p

p if Dpi (a) /∈ K p.

It is enough to show thatI is a Hasse ideal, since it is clearly maximal and we can setL := K Xn/I (note thata1/p ∈ L andL is generated overK by elements fromK 1/p, soL p ⊆ K ).

We will use formula (2.2) from the proof of [15, 2.4] several times. It says that fori < pn−1 anda ∈ K , Dip(ap) = Di (a)p. Actually, it is formulated for Hasse fields but istrue in the truncated case as well.

We need to apply the Hasse derivation to the generators only. Let

0 i < pn−1, 0 j < n − 1, 0 < l < pn − 1, 0 k < (p − 1)pn−1,

0 < m < pn.

We have

Dl (X(pn−1+k)) = cl ,pn−1+k X(l+pn−1+k).


In the casepi = p, we have

D1((X(i ))p − Dpi (a)) = −Dpi D1(a) = 0,

Dpj +1((X(i ))p − Dpi (a)) = Dpj (X(i ))p − Dpj +1 Dpi (a)

= cppj ,i

X(pj +i )p − cpj +1,pi Dp(pj +i )(a).

Now we want to check that ifpj + i pn−1, then cpj ,i = cpj +1,pi = 0, and that

l + pn−1 + k pn implies cl ,pn−1+k = 0. But this clearly follows from the fact thatα, β < pγ , α + β pγ impliescα,β = 0.To finish this case we need to notice that for allk, l

kp ≡ k (mod p), ck,l ≡ cpk,pl (mod p).

In the casepi = 1 we compute

Dm(X(i ) − Dpi (a)1/p)p = Dm(X(i ))p − Dm(Dpi (a)1/p)p

= cpi,m(X(i+m))p − DpmDpi (a) = cp

i,m(X(i+m))p − cpm,pi Dp(i+m)(a)

= ci,m[(X(i+m))p − Dp(i+m)(a)],since

cpi,m = ci,m = cpi,pm = cpm,pi .

Hence ifci,m = 0 (e.g. if i + m pn), we get

Dm(X(i ) − Dpi (a)1/p)p = 0 = Dm(X(i ) − Dpi (a)1/p),

and if ci,m = 0, then

Dp(i+m)(a) = (1/ci,m)Dm(Dpi (a)1/p)p ∈ K p.

Thereforepi+m = 1 and (since the Frobenius map is injective)

Dm(X(i ) − Dpi (a)1/p) = ci,m(X(i+m) − Dp(i+m)(a)1/p) ∈ I .

In the case of severaln-truncated Hasse derivations the proof goes through using1.1(i).

Note that the field extension we have constructed in the previous fact is purelyinseparable and smallest, but an extension of the Hasse derivation may be defined in manyways (it becomes evident in the proof of the next lemma). In the case of a Hasse fieldDpn−1(a1/p) = Dpn(a)1/p for eachn, which gives the smallest strict extension of a Hassefield as in [15, 2.4].

Lemma 2.6. The class of n-truncated Hasse fields does not have the amalgamationproperty.

Proof. Note that in the idealI (in the proof of2.5), X(pn−1) could have been replaced byX(pn−1) − b, for any absolute constantb. Therefore the value ofDpn−1(a1/p) can be anyabsolute constant and we easily get two extensions which cannot be amalgamated.


Lemma 2.7. Each strict n-truncated Hasse field(K , D1,<pn , . . . , De,<pn) embeds into astrict n-truncated Hasse field of inseparability degree e.

Proof. Let e′ be the inseparability degree ofK . By [15, 2.1],e′ e. Let e = e′ + f .By [15, 2.1] and [16, 2], we can assume that there isb1, . . . , be′ ⊂ K , a canonical

p-basis of depth 1 of(K , D1,<pn , . . . , De′,<pn).If f = 0, there is nothing to do. Letf > 0 and consider the field of rational

functions L = K (X1, . . . , X f ). By formula (2.1) in the proof of [7, 2.3], there areD′

1,<pn, . . . , D′e,<pn , Hasse derivations onL extendingD1,<pn, . . . , De,<pn respectively,

such that

D′i,1(X j ) =

1 if i = j + e′0 otherwise.

We want to check that(L, D′1,<pn , . . . , D′

e,<pn) is ann-truncated Hasse field. We havealready noticed inSection 1that we only need to check the iterativity and commutativityconditions on a set of ring generators ofL. By the uniqueness clause in [7, 2.3] (or formula(2.1) there), we need to check only field generators.

Let 1 i , i ′ e, 1 j , j ′ < pn and 1 k f . We have

Di, j Di ′ , j ′(Xk) = 0 = Di ′ , j ′ Di, j (Xk),

Di, j Di, j ′ (Xk) = 0 =

0 if j + j ′ pn

Di, j + j ′(Xk) if j + j ′ < pn.

Therefore(L, D′1,<pn, . . . , D′

e,<pn) is ann-truncated Hasse field.Note that B := b1, . . . , be′ , X1, . . . , X f is a p-basis of L (in particular L has

inseparability degreee).Takea ∈ L \ L p. We need to show thata is not a constant.

Case 1: a ∈ L p(X1, . . . , X f ).By 2.2, constants of(L p(X1, . . . , X f ), D′

e′+1,<pn, . . . , D′e,<pn) coincide withL p, so there

is i f such thatD′e′+i,1(a) = 0.

Case 2: a /∈ L p(X1, . . . , X f ).By 2.2, L p(X1, . . . , X f ) coincides with the constants of(L, D′

1,<pn , . . . , D′e′,<pn).

Thereforea is not a constant.

Lemma 2.8. Any n-truncated Hasse field structure extends uniquely to the separableclosure.

Proof. We do it first for onen-truncated Hasse derivation (not necessarily iterative)D ona field K . For anyi < n, a ∈ K and f ∈ K [X] we have

Dpi ( f (a)) = f Dpi (a) + f ′(a)Dpi (a) + w,

where in f Dpi each coefficient fromf is hit by Dpi and inw only D<pi appears. Thisimplies the uniqueness of the extension to the separable closure. The existence can beproved in the same way as in2.5, after unravelling the formula forw.

Commutativity and iterativity assertions follow from the uniqueness as in the proofof [15, 2.3].


One can also see that the proof of the Hasse field version of2.8(see [7, 2.5]) works alsoin the truncated case.

Fact 2.9. Strict n-truncated Hasse fields have the amalgamation property.

Proof. As in [15, 2.6], the amalgamation property follows from the uniqueness ofextensions ofn-truncated Hasse derivations to the separable closure (2.8) and the fact thatan n-truncated Hasse field extension of a strictn-truncated Hasse field is separable [15,2.1].

Theorem 2.10. Let Kn = (K , D1,<pn , . . . , De,<pn) be an n-truncated Hasse field. Thefollowing are equivalent.

(i) Kn is existentially closed.(ii) Kn is separably closed, strict and of inseparability degree e.(iii) Kn is a reduct of an existentially closed Hasse field.

Therefore the theory of n-truncated Hasse fields has a model companion.

Proof. (i)−→(ii). Suppose thatKn is existentially closed. By2.5, Kn is strict. By [15, 2.1]anyn-truncated Hasse field extension ofKn has the inseparability degree bounded bye. By2.2, having inseparability degreee is an existential condition, soKn has the inseparabilitydegreee by 2.7. By 2.8, Kn is separably closed.(ii)−→(iii). Assume thatKn is separably closed, strict and of inseparability degreee. By2.4, Kn has a canonicalp-basis of depthn. We use thatp-basis to expandKn to a Hassefield as in [15, 4.1]. By [15, 1.1], the Hasse field we have obtained is existentially closed.(iii) −→(i). SupposeKn is a reduct ofK = (K , D1, . . . , De), an existentially closed Hassefield. TakeLn, ann-truncated Hasse field extension ofKn. It is enough to embedLn intoMn, the n-truncated reduct of a Hasse fieldM extendingK. We can assume thatLn isfinitely generated (as a field) overK .

Take K′, an ω-saturated elementary extension ofK. By 2.9, K′n and Ln can be

amalgamated overKn into ann-truncated Hasse fieldMn. By [15, 4.2],K′ has a canonicalp-basis. SinceLn is finitely generated overK , we can assume thatMn is finitely generatedoverK′. By 2.3, Mn can be expanded to a Hasse fieldM extendingK′ (so extendingK,too) .

Let SCHp,e,n denote the theory of existentially closedn-truncated Hasse fields, andLn,e denote the language ofn-truncated Hasse fields. By2.10, the axioms of SCHp,e,n

are exactly the axioms of SCHp,e restricted toLn,e. As in [15], SCHp,e,n is stablenot superstable, since after choosing a canonicalp-basis any model of SCHp,e,n is bi-interpretable with the underlying separably closed field.

Note that SCHp,1,1 coincides with 1-DCF, the Wood theory from [14], which is thetheory of existentially closed differential fields with nilpotent derivations of orderp.Wood considers alson-DCF theories, where derivations have orderpn. Higher order (non-iterative) analogues ofn-DCF were developed and studied in [6].

Fact 2.11. SCHp,e,n does not have quantifier elimination in Ln,e and has quantifierelimination in Ln,e ∪ p-th root function.


Proof. The first part follows from2.6.For the second part, it is enough to see that strictness is a universal property with the

p-th root function, so quantifier elimination follows from2.9.

Note that the results of this section would be trivial if it was possible to expand any trun-cated Hasse field to a Hasse field. This is not the case though; see Example 2.2.3 in [12].

3. Geometric axioms for existentially closed Hasse fields (one derivation)

The results of this section form a proper subset of the results of the next section.However, we have the feeling that in the case of several Hasse derivations the presentationis somewhat blurred by a multiple usage of multi-indices. Therefore, we prefer to start withthe case of a single Hasse derivation, and then just to point out how to generalize the resultsto the case of several Hasse derivations.

Our axioms involve prolongation spaces, and we start by recalling the necessarydefinitions. In [11], the general framework for prolongation spaces with respect to anarbitrary operator is given. Let(K , D) be a fixed Hasse field with a single Hasse derivation.We assume that all the fields we consider are embedded into a big (pure) algebraicallyclosed fieldΩ .

We identify algebraic varieties withΩ -points of algebraic varieties definable overΩ .For exampleAm = Am(Ω) = Ωm is anm-affine space.

Let V be an affine variety definable overK and IK (V) ⊂ K [X] be the zero-ideal ofV over K . We consider now ann-truncated Hasse derivation onK Xn as a sequence ofmaps(Di : K [X] → K Xn)i<pn .

Definition 3.1. Then-th prolongation ofV , denoted by∇n(V), is an affine variety givenby the ideal(D<pn(IK (V))) ⊂ K Xn.

The above definition differs from the general definition from [11] just by twisting theindices. Usually,∇n(V) denotes the zeros ofD<n(IK (V)). Since we are going to consideronly the prolongation spaces corresponding to ideals of the formD<pn(IK (V)), we adoptthe more convenient notation.

Fact 3.2. Let(K , D) ⊂ (L, D′) be a Hasse field extension, V be an affine variety definableover K and a∈ V(L). Then for each n,D′

<pn(a) ∈ ∇n(V).

Proof. It is enough to see that for eachf ∈ K [X], d ∈ L, we have

D′<pn( f (d)) = D<pn( f )(D′

<pn(d)).

For a variety V definedover K (i.e. IK (V) generatesIΩ (V)), ∇nV(K ) naturallycorresponds toV(K [X]/(X pn

)), where theK -algebra structure onK [X]/(X pn) is given

by a homomorphism

K a →∑i<pn

Di (a)Xi ∈ K [X]/(X pn).

Therefore, on the level of coordinate rings,∇n is a left-adjoint functor to the functor takinga K -algebraR to R[X]/(X pn

), where theK -algebra structure onR[X]/(X pn) comes


from the above homomorphism composed with the natural embeddingK [X]/(X pn) ⊂

R[X]/(X pn).

The prolongation spaces are closely related to the arc spaces (see [4]), which correspondto V(K [X]/(X pn

)), whereK [X]/(X pn) has the usualK -algebra structure (corresponding

to the 0-Hasse derivation). In particular,∇1(V) projects onto a torsor of the tangent spaceto V , and∇1(V) projects onto the tangent space toV if V is defined over the constants (inthe standard notation, forV defined over the constants,∇1(V) is exactly the tangent spaceof V ).

Let us fix b = (b0, . . . , bpn−1) ∈ Ωmpnand setV = locusK (b0), W = locusK (b).

Recall that for a tuplec ∈ Ω l , locusK (c) is an affine algebraic subvariety (not necessarilyirreducible) ofΩ l defined byIK (c) := f ∈ K [X] | f (c) = 0.

We are interested in finding geometric conditions (i.e. in terms ofV and W) forextending(K , D<pn) to ann-truncated Hasse field(K (b), D′

<pn) such thatD′<pn(b0) = b.

For the proof of the next lemma we need the notion of ann-truncated Hasse derivationof a homomorphism of rings. It is quite natural, for a ring homomorphismf : R → S, thata sequence of maps(Di : R → S)i<pn is ann-truncated Hasse derivation of fif D0 = fand the following map:

R a →pn−1∑i=0

Di (a)Xi ∈ S[X]/(X pn)

is a ring homomorphism. As an example we havef : K [X] → K Xn the naturalembedding andD<pn the restrictions of then-truncated Hasse derivations onK Xn. Notethat if R = S, then it does not make much sense to talk about an iterativen-truncated Hassederivation of f .

Lemma 3.3. If b ⊂ K (b0) and b ∈ ∇n(V), then there isD′<pn , an n-truncated

Hasse derivation (not necessarily iterative) on K(b0) such thatD′<pn extendsD<pn and

D′<pn(b0) = b.

Proof. It is clear that

b ∈ ∇n(V) if and only if D<pn(IK (b0))(b) = 0.

Therefore, the naturaln-truncated Hasse derivation of the ring embeddingK [X] ⊂ K Xn

factors through

K [b0] = K [X]/IK (b0) → K Xn/IK (b) = K [b]if and only if b ∈ ∇n(V). Hence we get ann-truncated Hasse derivation of the ringembeddingK [b0] ⊂ K [b].

By the formula (2.1) in the proof of [7, 2.3], anyn-truncated Hasse derivation of anembedding of domains extends to ann-truncated Hasse derivation of the correspondingembedding of their fields of fractions.

The next fact is crucial for giving geometric axioms for SCHp,e. It is similar to [5,1.7] which was motivated by [9]. However there is an additional condition in [5, 1.7](corresponding to the separant condition in [13]) which is not present in [9] because of the


triviality of separability issues in the characteristic 0 case. A priori, our case here is evenmore complicated, since we also have to care about the iterativity condition. However, itturns out that the geometric iterativity condition already implies the additional assumptionfrom [5, 1.7] and even the conditionW ⊂ ∇n(V).

Fact 3.4. The following are equivalent:

(i) There is an n-truncated Hasse field extension(K , D<pn) ⊂ (K (b), D′<pn) such that

D′<pn(b0) = b.

(ii) There is an n-truncated Hasse field extension(K , D<pn) ⊂ (L, D′<pn) such that

b ⊂ L andD′<pn(b0) = b.

(iii) cn(W) ⊂ ∇n(W).

Proof. (i)−→(ii). Obvious.(ii)−→(iii). By 1.1(ii),

cn(b) = cnD′<pn(b0) = D′

<pnD′<pn(b0) = D′

<pn(b).

Thereforecn(W) ⊂ ∇n(W) (by 3.2).(iii) −→ (i). Since cn(b) ⊂ K (b), we get by3.3 (with b playing the role ofb0) ann-truncated Hasse derivation onK (b) such that D′

<pn(b) = cn(b). In particularD′

<pn(b0) = b.D′

<pn is iterative onb0; therefore (by1.1(iii)) D′<pn is iterative onK (b).

We can give now the geometric axioms for existentially closed fields with one Hassederivation.

Geometric axioms for SCHp,1

For any natural numbersn, m, suppose thatV ⊂ Am, W ⊂ Ampnare K -irreducible

K -varieties, andT is a properK -subvariety ofW. If W projects generically ontoV ,andcn(W) ⊂ ∇nW, then there is ana ∈ V(K ) such thatD<pn(a) ∈ W \ T .

The proof of the next theorem is almost the same as the proof of [5, 2.1]. The onlydifference comes from the fact that we cannot determine a Hasse field extension by meansof finite data (which can be governed by a formula) as in the differential case. That couldlead to serious trouble, but thanks toTheorem 2.10the proof goes through.

Theorem 3.5. (K , D) is an existentially closed Hasse field with one Hasse derivation ifand only if(K , D) is a model of the geometric axioms forSCHp,1.

Proof. =⇒ TakeV , W andT satisfying the axiom assumptions. SinceV andW areK -irreducibleK -varieties andW projects generically onV , there areb0, b ⊂ Ω such thatV = locusK (b0), W = locusK (b) andb = (b0, . . . , bpn−1).

By 3.4, there exists ann-truncated Hasse derivationD′<pn on K (b) such that

D′<pn(b0) = b. Obviouslyb ∈ W \ T . By 2.10, (K , D<pn ) is existentially closed as

ann-truncated Hasse field; hence we can findb′0 ∈ V(K ) such thatD<pn(b′

0) ∈ W \ T .


⇐= We could check that models of our axioms are strict separably closed Hasse fieldsof inseparability degree 1, and quote [15, 1.1]. Instead, we repeat an argument from [5,2.1].

By axioms for Hasse fields, it is enough to show that forx = (x0, . . . , xpn−1)

and ζ(x) any quantifier-free formula overK in the language of fields, if(M, D′) |=(∃x0)ζ(D′

<pn(x0)) for some Hasse field(M, D′) extending(K , D), then (K , D) |=(∃x0)ζ(D<pn(x0)).

Take b0 ⊂ M such that(M, D′) |= ζ(D′<pn(b0)) and let V = locusK (b0), W =

locusK (b) for b = D′<pn(b0). We takeT such that the axiom assumptions concerning it

are satisfied andζ(x) is implied by “x ∈ W \ T”.By 3.4, the axiom assumptions hold. Hence we geta0 ∈ V(K ) such thatD<pn(a0) ∈

W \ X. Therefore(K , D) |= (∃x0)ζ(D<pn(x0)).

Coming back to our original motivation, the equationD1(X) = X corresponds to thefollowing situation (assume for convenience thatp = 2):

V = A1, W = Z(X1 − X2) = (x, x) : x ∈ A

1 ⊂ A2 = ∇1(V),

c1(W) = (x, x, x, 0) : x ∈ A1 ⊂ A

4 = ∇1(∇1(V)),

∇1(W) = Z(X1 − X2, X′1 − X′

2) = (x, x, x′, x′) : x, x′ ∈ A1.

By 3.5, we can not find a generic point ofW in W ∩ graph(D1) (corresponding to ageneric solution ofD1X = X), sincec1(W) is not a subset of∇1(W).

The axioms above can be used to find solutions of certain linear Hasse differentialequations as in Section 6 of [10]. However such solutions can be also easily found justby using that the Hasse field in question is existentially closed and not caring about theparticular axioms.

It is easy to see that for a natural numberN, the above geometric axioms forn Ngive us axioms for SCHp,1,N .

One could wonder whether the theory of non-iterative (n-truncated) Hasse fields iscompanionable. In my previous attempt, I was trying to find those axioms first, and thenadd the geometric iterativity condition. It did not work though. Of course, forn = 1 andp = 2, the non-iterative theory has a model companion which coincides with DCFp [13].In the more general case ofe derivations (stilln = 1 andp = 2), the non-iterative theoryhas a model companion which coincides with DCFe

p (see [8, 2.1]).

Question 1. Is the non-iterative theory of (n-truncated) Hasse fields companionable?

4. Geometric axioms for existentially closed Hasse fields (several derivations)

In this section we observe that after suitable generalizations, the results of the previoussection hold for Hasse fields with several derivations. Almost everything is fairly straight-forward; only forcing commutativity in the generalization of3.4causes some problems.

We fix (K , D1, . . . , De), a Hasse field. Let us denote0, 1, . . . , pn − 1 by [pn]. Fori = (i1, . . . , i e) ∈ [pn]×e, we define

D<pn := (Di)i∈[pn]×e.


For an affine varietyV definable overK , and forI ⊂ 1, . . . , e, let

∇I ,n(V) := Z((Di(IK (V)))i∈[pn]I ).

For example ifI = 1, 2, e, then in the definition of∇I ,n only D1, D2, De matter. We alsodefine

∇n(V) := ∇1,...,e,n(V) , ∇i,n(V) := ∇i ,n(V).

For eachn there is a linear morphism

cn : Apne → A

p2ne, cn((ai)i∈[pn]×e) = (ci,jai+j)i,j∈[pn]×e,

whereci,jai+j makes sense in the same way as inSection 1.Again we fix b = (bi)i∈[pn]×e ∈ Ωmpne

and setV = locusK (b0), W = locusK (b)

(b0 := b(0,...,0)). We are interested in finding geometric conditions (i.e. in termsof V and W) for extending (K , D1,<pn , . . . , De,<pn) to an n-truncated Hasse field(K (b), D′

1,<pn, . . . , D′e,<pn) such thatD′

<pn(b0) = b.

Lemma 4.1. If b ⊂ K (b0) and b∈ ∇n(V), then there are commuting n-truncated Hassederivations (not necessarily iterative)D′

1,<pn, . . . , D′e,<pn on K(b0) such that for each

i e,D′i,<pn extendsDi,<pn andD′

<pn(b0) = b.

Proof. For I a subset of1, . . . , e, let

bI := (bi)i∈[pn]I .

Let us fix for a momentI ⊂ 1, . . . , e andi ∈ 1, . . . , e \ I .Since∇ j ,n and∇k,n commute for anyj , k e, we get

∇i,n(∇I ,n(V)) = ∇I ∪i ,n(V).

SinceK [bI ∪i ] ⊂ K [b] ⊂ K (b0) ⊂ K (bI ), we get (by3.4) DIi,<pn , ann-truncated Hasse

derivation onK (b0) extendingDi,<pn and such that

DIi,<pn (bI ) = bI ∪i .

In particular,DIi,<pn (b0) = bi ; hence alln-truncated Hasse derivations of the formDI

i,<pn

(for fixed i and anyI ⊂ 1, . . . , e \ i ) coincide. Let us denote thisn-truncated Hassederivation byD′

i,<pn . We get, for eachi , j e,

D′i,<pn D′

j ,<pn(b0) = D′i,<pn (b j ) = bi, j = D′

j ,<pn(bi ) = D′j ,<pnD′

i,<pn (b0).

Hence then-truncated Hasse derivationsD′i,<pn , D′

j ,<pn commute on the setK ∪ b0which generatesK (b0) as a field; therefore they commute everywhere.

Using4.1, the next fact can be proved in the same way as3.4.

Fact 4.2. The following are equivalent:

(i) There is an n-truncated Hasse field extension(K , D1,<pn , . . . , De,<pn) ⊂(K (b), D′

1,<pn, . . . , D′e,<pn) such thatD′

<pn(b0) = b.


(ii) There is an n-truncated Hasse field extension(K , D1,<pn , . . . , De,<pn) ⊂(L, D′

1,<pn , . . . , D′e,<pn) such that b⊂ L andD′

<pn(b0) = b.

(iii) cn(W) ⊂ ∇n(W).

The reader may become suspicious, because he or she may have a feeling that we didnot add any geometric commutativity conditions. But such conditions are hidden in thedefinition of the mapcn. The matrix definingcn is symmetric; in particular we have

c(i,0),(0, j )a(i,0)+(0, j ) = c( j ,0),(0,i )a( j ,0)+(0,i ).

One could still doubt why do we get commutativity inLemma 4.1, since there was no mapcn there. This is because the iterations between differentn-truncated Hasse derivations arehidden in the definition of∇n and also different∇i,n and∇ j ,n commute.

We continue as inSection 3.

Geometric axioms for SCHp,e

For any natural numbersn, m, suppose thatV ⊂ Am, W ⊂ Ampneare K -irreducible

K -varieties, andZ is a properK -subvariety ofW. If W projects generically ontoV ,andcn(W) ⊂ ∇nW, then there is ana ∈ V(K ) such thatD<pn(a) ∈ W \ Z.

The next theorem can be proved in the same way asTheorem 3.5.

Theorem 4.3. (K , D1, . . . , De) is an existentially closed Hasse field if and only if(K , D1, . . . , De) is a model of the geometric axioms forSCHp,e.

Obviously, forn N, the above axioms still axiomatize SCHp,e,N .

Acknowledgement

The author was supported by a postdoc under NSF Focused Research Grant DMS 01-00979 and the Polish KBN grant 2 P03A 018 24.

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geometric axioms for existentially closed hasse fields

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