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valued fields established by Kuhlmann in [Kuh16] and the fact that finite extensions of Fp((t)) inside Q Fp((t )) are isomorphic to Fp((t)) itself. Q This important connection between the first order theories of Fp((t)) and Fp((t )) in the language Q of valued fields served as a motivation for looking more closely at the first order theory of Fp((t )), and more generally of equal characteristic tame fields, in Lt, the language of valued fields with a distinguished constant symbol for t. The main result of this paper is the following.

Theorem 1. Let F be a perfect field of characteristic p which is decidable in the language of rings and let Γ be a p-divisible decidable ordered group which is decidable in the language of ordered groups with Γ a distinguished constant symbol 1. Then, F((t )) is decidable in Lt, the language of valued fields with a distinguished constant symbol for t.

The language Lt is needed to establish a complete analogue to decidability results of extensions of

Qp in the language of rings, since Fp[t] in Fp((t)) is like Z in Qp. Further motivating this analogy, Kartas

has recently showed that decidability results of equal characteristic fields in Lt can be transferred to decidability results for mixed characteristic fields in the language of valued fields [Kar20]. Hence, the results in this paper can be used to obtain decidability results for tame fields in mixed characterstic, which was previously unknown and has seemed to be inaccessible working in the language of rings.

2 Preliminaries

2.1 Notation and basic algebraic preliminaries

For a valued field (K, v), we will denote its value group by vK and its residue field by Kv. For an element γ ∈ vK, we write

K≥γ = {x ∈ K | v(x) ≥ γ}

and

K<γ = {x ∈ K | v(x) <γ}.

In particular, the valuation ring of K is written as K≥0. A valued field is called (algebraically) maximal if it has no proper immediate (algebraic) extension. For a field F, we denote by F¯ its algebraic closure. If Γ is an ordered abelian group, we denote by F((tΓ )) the Hahn field, or the field of generalised power series, consisting of formal expressions on the form γ x = X aγt γ∈Γ with well-ordered support. γ For a generalised power series x = Pγ∈Γ aγt , we will interchangeably use the notations

γ x = X aγt γ≥γ0

and

γi x = X ait i∈I

2 where γ0 is minimal such that aγ0 6=0 and I is a well-ordered index set. For r ∈ Γ , we write

γ x

γ x≤r = X aγt . γ∈Γ≤r

γi Similarly, with the notation x = Pi∈I ait , we write xpolynomial X − X − t , namely Pn≥0 t [Abh56]. A standard result (see for example Theorem 1 in [Poo93]) is that F((tΓ )) is maximal. It follows that F((tΓ )) algebraically closed if F is algebraically closed and Γ is divisible [Poo93, Corollary 4]. Throughout, Γ we will assume that t ∈ F((t )). More precisely, this amounts to choosing a positive element γ0 ∈ Γ and defining t = tγ0 .

Lemma 2.1.1. Let F be a field of characteristic p and let K be the relative algebraic closure of Fp(t) in F((tΓ )). Then K is contained in Kv((tvK )).

Q Γ Γ Q Proof. Consider F¯p((t )) and F((t )) as subfields of the algebraic closure of F((t )). Since F¯p((t )) is

algebraically closed and contains Fp(t), we have that K is a subfield of

Q Γ G F¯p((t )) ∩ F((t )) = E((t )),

where E = F¯p ∩ F and G is the relative divisible hull of v(t) in Γ . Since E is contained in Kv, we thus get that E = Kv. The result follows since vK = G.

Throughout, we will denote by p an arbitrary fixed prime number. A polynomial P is called additive if P (X + Y )= P (X)+ P (Y ) as polynomials in X and Y . To emphasise effectiveness, we include a proof of the following standard result.

Lemma 2.1.2 (Ore’s Lemma). Let K be a field of characteristic p and let f(X) ∈ K[X]. Then there is an additive polynomial P (X) ∈ K[X] such that f divides P . Furthermore, P is effective.

Proof. By writing deg(f) pi k X ≡ X bk,iX mod f(X) k=0

for 0 ≤ i ≤ deg(f), using for example Euclid’s algorithm, we can find ak such that

deg(f) pk X akX ≡ 0 mod f(x). k=0

2.2 Model theory and decidability

While model theoretic terminology in this paper is sparse, when we use it we mostly follow the conventions in [TZ12]. If L is a language and A and B are L-structures, we write A ≡ B if A and B have the same L- theories. Furthermore, if C is a common subset of A and B, we write A ≡C B if A and B have the same L(C)-theories, where L(C) denotes the language L extended by adjoining constant symbols for the elements in C, interpreted in the natural way in A and B.

3 We denote by Lring = {+, −, ·, 0, 1} the language of fields and by Log = {+, <, 0} the language of −1 ordered groups. We extend Lring to the language of valued fields Lval = {+, −, ·, , 0, 1, div}, where div is a binary relation symbol. For a valued field (K, v) and elements a,b ∈ K, the relation div(a,b) is interpreted as v(a) ≥ v(b). When we talk about the theories of Kv and vK, we mean the Lring-theory and Log-theory respectively, if not stated otherwise. Γ We let Lt = Lval∪{t}, where t is a constant symbol not in Lval. A Hahn field F((t )) is an Lt-structure by our general assumption that t ∈ F((tΓ )) and by considering the t-adic valuation on F((tΓ )). With an algorithm or a decision procedure, we mean a Turing machine, or any other equivalent model of computation. A first order theory T in a language L is recursively enumerable if there is an algorithm which lists T . More precisely, this algorithm takes L-sentences as inputs and returns TRUE if the sentence is in T . We then say that this algorithm enumerates T . If T is enumerated by an algorithm which in addition returns FALSE if φ is not in T , then T is called decidable. We will use the fact that a recursively enumerable complete theory is decidable, which is seen by using the algorithm which enumerates T to determine which one of φ and ¬φ are in T . If T has a recursively enumerable axiomatisation, then T is recursively enumerable, by listing the (finitely many) sentences of length at most n that one can deduce from the first n axioms of T , and then repeat for n +1 and so on.

Using results on the Log-theories of ordered abelian groups in [RZ60], we get the following lemma. L 1 Lemma 2.2.1. The og(1)-theories of p∞ Z and Q are decidable. L+ L Proof. Let og be the language obtained by adding to og predicates Dn for each n coprime to p. These predicates are intepreted as

Dn(x) ⇔ ∃y(ny = x).

L+ 1 From [RZ60], we have recursively enumerable axiomatisations the og-theories of p∞ Z and Q, which are also shown to be model complete. Call these theories T1/p∞ and TQ respectively. In particular, all L+ models of T1/p∞ are p-divisible and all models of TQ are divisible. We now claim that the og(1)-theories 1 of p∞ Z and Q are given by

T1/p∞ ∪{1 > 0} ∪ ∀X(nX 6= 1) | n ∈ N \ pN

and

TQ ∪{1 > 0}

1 respectively. Indeed, any model of these extended theories contain p∞ Z respectively Q as submodels. By L+ L+ model completeness of the og-theories, these are elementary substructures as og-structures. Hence, L+ they are also elementary substructures as og(1)-structures. In other words, they are prime models of L+ the respective og(1)-theories, and so the theories are complete. Since we added a recursively enumerable L+ L+ set of sentences to the recursively enumerable og-theories, we have that the og(1)-theories are also ( recursively enumerable, hence decidable. Since the Log1)-theories are subsets of these theories, the result follows.

2.3 Kaplansky fields

For a valued field (K, v) with char(Kv) = p, the following two conditions constitutes what Kaplansky calls Hypothesis A [Kap42], which provides a criterion for understanding all the immediate extensions of K.

1. Any non-zero additive polynomial f ∈ Kv[X] is surjective on Kv.

2. The value group vK is p-divisible.

4 A valued field satisfying Hypothesis A is also called a Kaplansky field. The importance of these conditions are captured in the following [Kap42, Theorem 5].

Theorem 2.3.1. Let (K, v) be a Kaplansky field. Then, (K, v) admits a unique maximal immediate extension (L, v), up to valuation preserving isomorphism over K.

The following result by Whaples gives an alternative characterisation of Kaplansky fields. Note that the original statement in Theorem 1 in [Wha57] refers to the residue field as a Kaplansky field, rather than the valued field itself.

Theorem 2.3.2. A valued field (K, v) is a Kaplansky field if and only if it satisfies the following.

1’. Kv has no algebraic extension of degree divisible by p.

1. The value group vK is p-divisible.

We will need the following results, which appear in Theorem 1.1 and Proposition 1.2 in [Kuh18].

Theorem 2.3.3. Let (L, v) be an algebraically maximal Kaplansky field. Then for any subfield K of L, we have that L contains a maximal immediate extension of K. Furthermore, if K is a relatively algebraically closed subfield of L, then (K, v) is also an algebraically maximal Kaplansky field.

2.4 Tame fields

When F is a perfect field and Γ is a p-divisible ordered abelian group, then F((tΓ )) falls in an elementary class of fields called tame fields. This class was studied extensively by Kuhlmann in [Kuh16] and we follow this approach.

Definition 2.4.1. An algebraic extension (L|K, v) of a henselian valued field (K, v) is called tame if every finite subextension E|K of L|K satisfies the following conditions.

1. (vE : vK) is prime to p;

2. Ev|Kv is separable;

3. E|K is defectless, i.e. [E : K] = (vE : vK)[Ev : Kv].

A tame field is a henselian valued field for which all algebraic extensions are tame.

That Hahn fields with perfect residue field and p-divisible value group are canonical examples of tame fields is, if not apparent from the definition, clear from the following alternative characterisations (see Theorem 3.2 and Corollary 3.3 in [Kuh16]).

Theorem 2.4.2. Let (K, v) be a henselian valued field and let p be the characteristic exponent of Kv. Then (K, v) is tame if and only of (K, v) is algebraically maximal, Kv is perfect and vK is p-divisible. If in addition K has characteristic exponent p, then (K, v) is tame if and only if it is algebraically maximal and perfect.

The following important result is Lemma 3.7 in [Kuh16].

Lemma 2.4.3. Let (L, v) be a tame field and let (K, v) ⊂ (L, v) be a relatively algebraically closed subfield. Suppose that Lv|Kv is an algebraic extension. Then (K, v) is a tame field, vL/vK is torsion free and Kv = Lv.

For our purposes, we note that the hypothesis in Lemma 2.4.3 can be weakened when working with Hahn fields to obtain a similar result.

5 Lemma 2.4.4. Let F be a perfect field of characteristic p and let Γ be a p-divisible value group. Let Γ (K, v) be the relative algebraic closure of Fp(t) in F((t )). Then (K, v) is a tame field and vL/vK is torsion free. Proof. By assumption on F and Γ , we have that Kv is perfect and vK is p-divisible. Hence, since Hahn fields are maximal, we have that Kv((tvK )) is tame. By Lemma 2.1.1 and Lemma 2.4.3, we thus get that (K, v) is tame. Since vK is the relative divisible hull of v(t) in Γ , we have that vL/vK is torsion free.

The following examples, inspired by Example 3.9 in [Kuh04] shows that the two previous lemmas are not vacuously true.

Example 2.4.5. Let s be transcendental over Fp(t) and let v be the t-adic valuation on Fp(s,t), so that 1/2 1/p∞ (Fp(s,t), v) has residue field Fp(s). Let L be a maximal immediate extension of Fp(t, (st) ) with the same valuation. Then L is a tame field by Theorem 2.4.2. Let K be the relative algebraic closure of 1 1 Fp(t) in L. Then vK = p∞ Z but vL = 2p∞ Z.

Similarly, tameness of the relative algebraic closure of Fp(t) can fail, as seen in this example.

Example 2.4.6. Again, let s be transcendental over Fp(t). We will consider all fields in this example as subfields of the Hahn field Q 1/p∞ M = F¯p((s ))((t ))

Q with the t-adic valuation v. To clarify the notation, we have that Mv = F¯p((s )). Note that M is a Kaplansky field. −1/pn Let α = Pn≥1 s and let E be the field given by adjoining

−1/pn β = α + X t n≥1

1/p∞ 1/p∞ p −1 −1 to Fp((s)) ((t)). Note that α∈ / Fp((s)) . Since β is a root of X − X − s − t , we have that 1/p∞ E/Fp((s)) ((t)) is an extension of degree p. Since v(β) = −1/p, adjoining β does not extend the 1/p∞ 1/p∞ residue field of Fp((s)) ((t)), i.e. Ev = Fp((s)) . By Theorem 2.3.3, we can let L be a maximal immediate extension of E inside M. Let K be the 1/p∞ relative algebraic closure of Fp(t) in L. Since Fp is relatively algebraically closed in Fp((s )), we have n F 1/p∞ 1 −1/p by Lemma 2.1.1 that K is contained in p((t )). Furthermore, since vK = p∞ , adding Pn≥1 t −1/pn gives an immediate algebraic extension of K. This is a proper extension, since if Pn≥1 t ∈ L, then α ∈ L. Hence, K is not algebraically maximal. One of the main results in [Kuh16] is that tame fields admits an Ax-Kochen Ershov principle in the language Lval. To mimic this result for Lt in Section 3.1, we will need the following. Definition 2.4.7. Let C be an elementary class of tame fields. If for every two fields (L, v), (F, w) ∈ C and every common defectless subfield (K, v) of (L, v) and (F, w) such that vL/vK is torsion free and

Lv|Kv is separable, the conditions vL ≡vK wF and Lv ≡Kv F w imply that (L, v) ≡(K,v) (F, w), then we will call C relatively subcomplete.

Theorem 2.4.8 (Theorem 7.1 in [Kuh16]). The class of tame fields is relatively subcomplete in the language Lval.

2.5 Finite automata

In this section, we include some standard results on finite automata following [AS03], as well as key results in [Ked06]. While this section does not contain any new findings, it is essential for our purposes

6 that the results on algebraicity of generalised power series obtained in [Ked06] are effective. We therefore add details to some of the proofs to emphasise this fact.

Definition 2.5.1. A deterministic finite automaton with output, or a DFAO, is a tuple M =

(Q,Σ,δ,q0, ∆, τ) where

• Q is a finite set (the states);

• Σ is a set (the input alphabet);

• δ is a function from Q × Σ to Q (the transition function);

• q0 ∈ Q (the initial state);

• ∆ is a finite set containing 0 (the output alphabet);

• τ is a function from Q to ∆ (the output function).

A deterministic finite automaton, or a DFA, is a DFAO with output alphabet {0, 1}. A nondeter- ministic finite automaton, or an NFA, is defined just as a DFA except for the transition function δ which is then a function from Q × Σ to the power set of Q. For a DFAO or an NFA, we say that a state q ∈ Q is an accepting state of M if τ(q) 6=0.

If M is a DFA with output function τ and ∆ is a set containing an element a, we denote by aM the DFAO given from M by replacing the output alphabet with ∆ and the output function with aτ (where multiplication by 0 and 1 behaves as expected). Given a set Σ, we denote by Σ∗ the set of finite strings with elements from Σ. If M is a DFAO with input alphabet Σ and transition function δ, we extend δ to the function

δ∗ : Q × Σ∗ → Q defined recursively by

δ∗(q, ∅)= q δ∗(q,wa)= δ(δ∗(q, w),a) where q ∈ Q, x ∈ Σ∗ and a ∈ Σ. Furthermore, we let

∗ fM : Σ → ∆ be the function defined by ∗ fM (w)= τ(δ (q0, w)).

Instead of giving the output function of an NFA, it is enough to specify its accepted states. We will thus often write an NFA N as N = (Q,Σ,δ,q0, F ), where F ⊂ Q are the accepted states of N. Note that any DFA is both a DFAO and an NFA, and that the following two definitions indeed agree on DFAs.

∗ Definition 2.5.2. Let M = (Q,Σ,δ,q0, ∆, τ) be a DFAO. For a string w ∈ Σ , we say that M accepts ∗ w if fM (w) 6=0. The set of strings in Σ accepted by M is called the language accepted by M.

∗ Definition 2.5.3. Let N = (Q,Σ,δ,q0, F ) be an NFA and let w = s1 ...sn ∈ Σ . An accepting path for w is a sequence of states q1,...,qn ∈ Q such that qi ∈ δ(qi−1,si) for i ∈{1,...,n} and qn ∈ F . We say that N accepts w if there exists an accepting path for w in N. The set of strings in Σ∗ accepted by N is called the language accepted by N.

7 As models of computation, an NFA is equivalent to a DFA in the following sense.

Theorem 2.5.4 ([AS03, Theorem 4.1.3]). Suppose that S is a language accepted by an NFA M. Then, S is also the language accepted by a DFA M ′. Furthermore, M ′ is computable from M.

∗ ∗ We denote by rev : Σ → Σ the function sending s1 ··· sn to sn ··· s1.

Theorem 2.5.5 ([AS03, Theorem 4.3.3]). Let M = (Q,Σ,δ,q0,τ,∆) be a DFAO with corresponding ′ ′ ′ ′ ′ ′ function fM . Then, there is a DFAO M = (Q ,Σ,δ , τ , q0, ∆ ) which is computable from M such that fM ′ = fM ◦ rev.

Lemma 2.5.6 ([Ked06, Lemma 2.2.2]). Fix a positive integer n. Let M = (Q,Σ,δ,q0, F ) be an NFA. For w ∈ Σ∗ the number of accepting paths for w in M, let a(w) be the number of accepting paths for w in M. Let

f : Σ∗ → Z/nZ w 7→ [a(w)].

′ ′ Then, there is a DFAO M such that f = fM ′ . Furthermore, M is computable from M.

We will mainly consider finite automata with input alphabet Σp = {0, 1,...,p − 1,.}.

∗ Definition 2.5.7. A string s = s1 ...sn ∈ Σp is said to be a valid base p-expansion if s1 6=0, sn 6=0 and sk is equal to the radix point for exactly on k ∈ {1,...,n}. If s is a valid base p-expansion and sk is the radix point, then we define the value of s to be

k−1 k−1−i k−i v(s)= X sip + X sip . i=1 i=k+1

Definition 2.5.8. Let M be a DFAO with input alphabet Σp. We say a state q ∈ Q is preradix (resp.

postradix) if there exists a valid base p expansion s = s1 ...sn accepted by M with sk equal to the radix ∗ point such that q = δ (q0,s1 ...si) for some i < k (resp. for some i ≥ k).

Remark 2.5.9. If M accepts a language consisting of only valid base p expansions, then no accepted state can be both preradix and postradix. Indeed, suppose that there is an accepted state q which is ′ ′ both preradix and postradix. Then, there are valid base p expansions s1 ...sm and s1 ...sn accepted by ′ M with radix points sk and sℓ respectively such that

∗ ∗ ′ ′ q = δ (q0,s1 ...si)= δ (q0,s1 ...sj ),

with i < k and j ≥ ℓ. We then have that

∗ ∗ ′ ′ δ (q0,s1 ...sm)= δ (q0,s1 ...sjsi+1 ...sm)

is an accepted state reached by a string with two radix points, which is a contradiction.

Definition 2.5.10. Let ∆ be a finite set. A function

1 f : N → ∆ p∞

1 is called p-automatic if there is a DFAO M = (Q,Σ,q0, ∆, τ) such that for any v ∈ p∞ Z, we have that f(v)= fM (s(v)).

8 The connection between finite automata and generalised power series is captured in the following result, which follows immediately from Theorem 4.1.3. in [Ked06].

1 Theorem 2.5.11. Let q be a power of a prime p and let f : p∞ N → Fq be a function with well-ordered support. Then the generalised power series

γ 1/p∞ X f(γ)t ∈ Fq[[t ]] 1 γ∈ p∞ N is algebraic over Fq(t) if and only if f is p-automatic.

It is often convenient to consider finite automata as edge-labeled directed graphs. In particular, this will be useful to verify effectiveness of results in this section.

Definition 2.5.12. Let M = (Q,Σ,δ,q0, ∆, τ) be a DFAO. The transition graph of M is the edge- labeled directed graph on the vertex set Q, with an edge from q ∈ Q to q′ ∈ Q labeled by s ∈ Σ if δ(q,s)= q′.

Definition 2.5.13. Let M = (Q,Σ,δ,q0, ∆, τ) be a DFAO. We say that a state q ∈ Q is reachable ∗ ∗ from a state q1 if there is some string s ∈ Σ such that δ (q1,s) = q. If q is reachable from q0, we say that q is reachable. If q is not reachable, we say that it is unreachable. If M has no unreachable states, we say that it is minimal.

Definition 2.5.14. A state q ∈ Q is said to be final if q is reachable and no other states than q are reachable from q. If there exists a final state reachable from q, we say that q is relevant. If q is not relevant, we say that it is irrelevant.

Remark 2.5.15. Given a DFAO M = (Q,Σ,δ,q0, ∆, τ), there is an effective procedure to determine the reachable states of Q. This can be seen intuitively by considering the transition graph of M and tracing backwards from each state. Hence, there is an effective procedure to obtain a minimal DFAO ′ ′ ′ M = (Q ,Σ,δ|Q′ , q0, ∆, τ|Q′ ) such that Q ⊂ Q consists of the reachable states in M simply by removing ′ all the unreachable states from Q. For this M , we have that fM ′ = fM

We will now consider a characterisation of transition graphs which will allow us to tell if fM is p-automatic with well-ordered support.

Definition 2.5.16. Let M = (Q,Σp, δ, q0, ∆, τ) be a DFAO. We say that M is well-formed (resp. 1 well-ordered) if there is an arbitrary (resp. a well-ordered) subset S ⊂ p∞ N such that the language accepted by M consists of the valid base p expansions of S. Similarly.

Remark 2.5.17. Given a DFAO M with input alphabet Σp, there is an effective procedure to check that it is well-formed from the transition graph G = (V, E) of M. Indeed, it is enough to verify the following.

• M is minimal (removing all the unreached states is effective).

• δ(q0, 0) is in a subset of V from which no accepted state is reachable.

• No accepted state is reached by 0.

• There are no preradix accepted states.

• There are no connections between postradix and preradix states, except instances of the radix point from the preradix to the postradix states.

9 • For any postradix state q, we have that the radix point sends q to q′, where δ(q′,s) = q′ for any ′ s ∈ Σp and τ(q )=0.

1 To verify that a DFA a language which gives base p expansions of a well-ordered subset of p∞ N we need a bit more.

Definition 2.5.18. A directed graph G = (V, E) with a distinguished vertex v ∈ V is called a rooted saguaro if the following hold.

1. Each vertex of G lies on at most one minimal cycle.

2. There exists directed paths from v to each vertex of G.

In this case, we say that v is a root of G. A minimal cycle of a rooted saguaro is called a lobe. An edge of a rooted saguaro is cyclic if it lies on a lobe and acyclic otherwise.

Definition 2.5.19. Let G = (V, E) be a rooted saguaro. A proper p-labeling of G is a function ℓ : E →{0,...,p − 1} with the following properties.

1. If v,w,x ∈ V , w 6= x and vw,vx ∈ E, then ℓ(vw) 6= ℓ(vx).

2. If v,w,x ∈ V , vw ∈ E lies on a lobe, then ℓ(vw) >ℓ(vx)

Theorem 2.5.20 ([Ked06, Theorem 7.1.6]). Let M be a DFAO with input alphabet Σp which is minimal and well-formed. For any state q of M, let Gq be the subgraphs of the transition graph consisting of relevant states reachable from q. Then M is well-ordered if and only if for each relevant postradix state q, we have that Gq is a rooted saguaro with root q, equipped with a proper p-labeling.

Remark 2.5.21. Since the transition graph of a DFAO is finite, there is an effective procedure to check if the conditions of Theorem 2.5.20 are met. We conclude that the set of well-ordered DFAOs with output alphabet Fq is decidable.

In Section 3.2, we will use Theorem 2.5.11 to determine which monic one variable polynomials over

Fp[t] have solutions in certain Hahn fields. To this end, we will need the following two lemmas describ- 1/p∞ ing an effective procedure for arithmetic of DFAOs representing elements in Fq[[t ]]. They appear implicitly in [Ked06] as Lemma 7.2.1 and Lemma 7.2.2 respectively. Again, the proofs are entirely due to Kedlaya and included to emphasise effectiveness.

1/p∞ Lemma 2.5.22. Let x, y ∈ Fq[[t ]] be p-automatic and that the corresponding DFAOs Mx and My are given. Then the DFAO Mx+y corresponding to x + y is computable from Mx and My.

Proof. Let Mx = (Qx,Σp,δx, q0,x, Fq, τx) and My = (Qy,Σp,δy, q0.y, Fq, τy). Define

˜ x y N = (Qx × Qy,Σp, δ, q0 , q0 , Fq, τ˜) where δ˜((qx, qy),s)= δx(qx,s)×δy(qy,s) and τ˜(qx, qy)= τx(qx)+τy(qy). By construction, N is a DFAO. ∗ We claim that fN (γ)= fMx (γ)+ fMy (γ), i.e. that x + y is given by N. Indeed, let w ∈ Σ be the base p expansion of γ. We have that

∗ fN (γ)=˜τ(δ˜ ((q0,x, q0,y), w)) ∗ ∗ =τ ˜(δx(q0, w),δy (q0,x, w)) ∗ ∗ = τx(δx(q0, w)) + τy(δy(q0,x, w))

= fMx (γ)+ fMy (γ).

Hence, N = Mx+y and we are done.

10 1/p∞ Lemma 2.5.23. Let x, y ∈ Fq[[t ]] be p-automatic and let Mx and My be the corresponding DFAOs.

Then the DFAO Mxy corresponding to xy is computable from Mx and My.

Proof. Let q r 1/p∞ x = X ai X t ∈ Fq[[t ]] i=1 r∈Si with ai ∈ Fq and ai 6= aj for i 6= j. Suppose that x corresponds to the DFAO M = (Q,Σp, δ, q0, Fq, τ). r We note that for every i, there is a computable DFAO Mi such that t corresponds to Mi. Pr∈Si Such an Mi is given by (Q,Σp, δ, q0, Fq, τi), where τi(q)=1 if τ(q) = ai and τi(q)=0 otherwise. By Lemma 2.5.22, we can therefore assume that x = tr and y = tr. For such x and y, the Pr∈Ax Pr∈Ay corresponding DFAOs Mx and My are in fact DFAs. By definition of multiplication in Hahn fields, the coefficient of a term with value γ in xy is equal to the number of ways to write r1 + r2 = γ with r1 ∈ Ax and r2 ∈ Ay. Since these coefficients are in characteristic p, we can take this number modulo p. We will see that this amounts to counting the number of accepting paths modulo p in a certain NFA ′ and then use Lemma 2.5.6 to conclude the result. To this end, let Mx = (Qx,Σp,δx, q0,x, Fq, τx) and ′ My = (Qx,Σp,δx, q0,x, Fq, τx) be the DFAs accepting the reversed languages of Mx and My respectively, ∗ ∗ as in Corollary 2.5.5. Let S be the subset of Σp × Σp consisting of pairs (w1, w2) with the following properties.

1. w1 and w2 have the same length.

2. w1 and w2 each end with 0.

3. w1 and w2 each have a single radix point, and both are in the same position.

4. After removing leading and trailing zeroes, w1 and w2 become the reversed valid base p expansions

of some i ∈ Ax and j ∈ Ay respectively.

We now construct a DFA M = (Q,Σp × Σp, δ, q0, {0, 1}, τ) such that S is the language accepted by M ∗ ∗ ∗ (note that we identify (Σp × Σp) with Σp × Σp ). Let R and A be distinguished elements and let

Q = (Qx × Qy) ⊔{R, A}.

Let q0 = (q0,x, q0,y). We define δ recursively as follows.

1. Let δ(q0, (0, 0)) = q0.

2. δ((q0,x, q), (0,s)) = (q0,x,δy(q,s)).

3. δ((q, q0,y), (s, 0)) = (δx(q,s), q0,y).

′ ′ ′ ′ ′ ′ 4. δ((q, q ), (s,s )) = (δx(q,s),δy(q ,s )) for all (q, q ) ∈ Qx ×Qy \{(q0,x, q0,y)} and all (s,s ) ∈ Σp ×Σp such that not only one of s and s′ are a radix point.

′ ′ 5. δ((q, q ), (., .)) = (δx(q, .),δy(q ,.)).

′ ′ 6. δ((q, q ), (0, 0)) = A if q ∈ Qx and q ∈ Qy are accepting states of Mx and My respectively.

7. δ(A, (0, 0)) = A.

8. δ(A, s)= R for all s ∈ Σp \{0}.

9. δ((q, q′), (s,s′)) = R if only one of s and s′ is a radix point.

10. δ(R,s)= R for all s ∈ Σp.

11 ′ ′ ′ Finally, we define τ(q, q )= τ(q)τ(q ) for all (q, q ) ∈ Qx × Qy, τ(A)=1 and τ(R)=0. Informally, the DFA M can be seen to accept the pairs of reversed base p expansions of the exponents

in Ax × Ay which are well set up for adding these pairs together under base p addition with carries. We ′ ′ ′ ′ ′ ′ will now construct an NFA M = (Q ,Σp,δ , q0, F ) which captures this addition. Let Q = Q ×{0, 1}, ′ ′ ′ let q0 = (q0, 0) and let F = F ×{0}, where F is the set of accepted states of M. To define δ , let (q,i) ∈ Q′ and consider the following cases.

1. If s ∈ {0,...,p − 1}, we include (q′, 0) (resp. (q′, 1)) in δ′((q,i),s) if there exists a pair (t,u) ∈ {0,...,p − 1}2 with t + u + i

2. If s is equal to the radix point, we include (q′,i) in δ((q,i),s) if δ(q, (s,s)) = q′, and we never include (q′, 1 − i).

∗ ′ Claim 1. Let w = s1 ...sm ∈ Σ . Then w is accepted by M if and only if there is (w1, w2) ∈ S

such that w is the sum of w1 and w2 under base p addition with carries.

Proof of Claim 1: Suppose that (w1, w2) ∈ S is such that w is the sum of w1 and w2 under base

p addition with carries. Since w1 and w2 both end with 0, they must have the same length as w. Let

q1,...,qm be an accepting path for (w1, w2) in M. Then, if ik is the carry at digit k in the base p ′ addition of w1 and w2, we have by definition of δ that (q1,i1),..., (qm,im) is an accepting path for w ′ in M . Note that im is necessarily 0, making (qm,im) an accepted state. For the converse, suppose that ′ ′ (q1,i1),..., (qm,im) is an accepting path for w in M . By definition of F , we then have that im =0 and

that qm is an accepted state in M. We will now recursively define

(w1, w2) = (t1 ...tm,u1 ...um) ∈ S

such that for each k ∈{0,...,m}, we have that s1 ...skik is the sum under base p addition with carries

of the k first symbols of w1 and w2. This shows in particular that w is the sum of w1 and w2 under base p addition with carries. The base case with k = 0 is vacuously true since this gives the empty

string. Supposing that the statement holds for k ∈{0,...,n − 1}, let (w1, w2) = (t1 ...tn,u1 ...un) ∈ S

be such that s1 ...skik is the sum under base p addition with carries of t1 ...tk and u1 ...uk. Let 2 ′ (tk+1,uk+1) ∈ Σp be as in the definition of δ such that δ(qk, (tk+1,uk+1)) = qk+1. The conditions ′ defining δ says exactly that skik+1 is the sum under base p addition with carries of tk+1, uk+1 and

ik. Hence, we get that s1 ...sk+1ik+1 is the sum under base p addition with carries of t1 ...tk+1 and ′ ′ 2 u1 ...uk+1. Furthermore, by definition of δ , for each ℓ ∈ {1,...,m}, there is a pair (s,s ) ∈ Σp such ′ that qℓ = δ(qℓ−1, (s,s )). Hence, q1,...,qm is an accepting path in M. Since S is the language accepted ′ ′ by M, we get that (t1 ...tk+1,u1 ...uk+1) is the initial string of some (w1, w2) ∈ S, and we are done proving the claim. ′ We conclude that the number of accepting paths of w in M is equal to the number of pairs (w1, w2) ∈ S which sum to w with its leading and trailing zeroes under ordinary base p addition with carries. By ′ Lemma 2.5.6, there is a DFAO N such that fN (w) equals the number of accepting paths of w in M modulo p.

A priori, there might be (w1, w2) ∈ S which sum to a reversed base p expansion of γ even though

(w1, w2) does not sum to w, since we need to take into account leading and trailing zeroes. Appending leading zeroes to w will account for more possible (w1, w2) ∈ S, and we will show that there is an effective bound to how many zeroes we need to append. To make this more precise, let m be greater than the number of states of M and suppose that w begins with m leading zeroes. We claim that for any pair

(w1, w2) ∈ S which sums to w, both w1 and w2 must begin with a leading zero. Indeed, suppose not. By choice of m, some state must be repeated within the first m digits when processing (w1, w2) ∈ S under

12 ∗ M (i.e. when computing δ (q0,u) for successive truncations u of (w1, w2)). Let (b1,b2) be the string that lead to the first arrival at the repeated state, let (u1,u2) be the string between the first and the second arrival, and let (e1,e2) be such that (w1, w2) = (b1u1e1,b2u2e2). Then

biei, biuiei, biuiuiei,..., represent the reversed base p expansions (with possible trailing zeroes but no leading zeroes) of some elements z0i,z1i,... of Ai. If bi is nonempty, then bi begins with a nonzero digit. If bi is empty, then ui begins with a nonzero digit. In both cases, ui is nonempty. Hence, the numbers z0i,z1i,... are all distinct. Furthermore, we will show that zj0 +zj1 = z for all j. This is true for j =1 is by assumption on

(w1, w2). By choice of bi and ui, we must have that bi and ui all have length less than or equal to m. Since w have m leading zeroes, we therefore get that in doing the addition, the stretch during which u1 and u2 are added begins with an incoming carry and ends with an outgoing carry, and that all digits produced before and during the stretch are zeroes. Thus, we may remove or repeat this stretch without changing the base p number represented by the sum. Since zj0 + zj1 = z for all j, one of the sequences z0i,z1i,... must be strictly decreasing, which is a contradiction. Hence, w1 and w2 both begin with leading zeroes.

In conclusion, if w has m leading zeroes, then (w1, w2) ∈ S sums to w if and only if (0w1, 0w2) ∈ S sums to 0w. This shows that number of accepting paths of such w equals the number of accepting paths of 1 0w. We conclude that the function that, given the reversed valid base p expansion of a number γ ∈ p∞ Z, computes the mod p reduction of the number of ways to write γ = i + j with (i, j) ∈ Ax × Ay is given by ′ ˜′ ˜′ ′ ′ ˜ ˜ a DFAO N = (Q ,Σp, δ , q˜0, Fp, τ˜ ) which is computable using N = (Q,Σp, δ, q˜0, Fp, τ˜) in the following ′ ′ ˜∗ way. Let G = (V, E) be the transition graph of N. We set the initial state of N to be q˜0 = δ (˜q0,m · 0), ′ ′ where m · 0 denotes the string with m zeroes. We let the transition graph of N be Gq˜0 . For any state q of N ′, we set τ˜′(q)=˜τ(δ˜(q, 0). In other words, N ′ amounts to appending m leading zeroes and one trailing zero to a string and then running the result through N. Finally, taking N ′ to be the reversal of N, as in Theorem 2.5.5 gives the desired result.

Remark 2.5.24. The results above show that, given a DFAO M and a polynomial f(X) over Fp[t], there is an effective procedure to determine if f(M)=0. We only need to note that a well-ordered 1/p∞ DFAO represents 0 ∈ Fq[[t ]] if and only if it has no accepted states and that there is an effective

procedure to construct a well-ordered DFAO which represents a ∈ Fq[t] (since such a DFAO only accepts finitely many strings).

3 Main results

3.1 An AKE-principle for tame fields in Lt

Throughout this section, let (L, v) be a tame field of residue characteristic p with p-divisible value group.

If L has characteristic p, let t ∈ L be transcendental over Fp with v(t) > 0 and let A = Fp(t). Otherwise, let t = p and let A = Q. Let (K, v) be the relative algebraic closure of A in L. Suppose furthermore that (K, v) is algebraically maximal and that vL/vK is torsion free. In particular, L can be an equal characteristic tame Hahn field, since K the conditions on K are then satisfied by Lemma 2.4.4.

We will use Theorem 2.4.8 to obtain an AKE-principle for the theory of (L, v) in the language Lt, following closely the proof of Lemma 6.1 in [Kuh16]. This AKE-principle was originally formulated for positive characteristic only. The general result was prompted by Konstantinos Kartas asking if this also holds in characteristic 0. Note that the constant symbol t is superfluous in this case. We keep it for uniformity of the proofs.

13 Remark 3.1.1. With the notation above, let (F, w) be a valued field containing A such that F w ≡

Lv. Then w|A is the t-adic valuation. Indeed, since char(F w) = p, we have that v(a)=0 for all a ∈ {1,...,p − 1}. Since w(t) > 0, the statement then follows from considering base t-expansions of elements in A.

Lemma 3.1.2. Let (F, w) be a tame field containing A and suppose that vL ≡v(t) wF and Lv ≡ F w.

Suppose furthermore that (K, v) is isomorphic over A to a subfield of (F, w). Then (L, v) ≡t (F, w).

Proof. By assumption, (K, v) is algebraically maximal. Since vL is p-divisible and vL/vK is torsion free,

we also have that vK is p-divisible. Since Kv is an algebraic extension of Fp, it is perfect. Hence, by Theorem 2.4.2, we have that (K, v) is tame. In particular, it is defectless. Since Lv|Kv is separable we

are in the situation of Definition 2.4.7 and it is enough to show that vL ≡vK wF and Lv ≡Kv F w. First, note that vA = hv(t)i and wA = hw(t)i, since v and w are the t-adic valuations on A. Since K is an algebraic extension of A, we therefore have that vK is a subgroup of the divisible hulls of hv(t)i and hw(t)i respectively. Hence, vK can be defined in the language of ordered abelian groups together with a constant symbol v(t), as an element mv(t)/n is the unique element x satisfying the formula nx = mv(t).

Since vL and wF are elementary equivalent in the language Lt and since v(t) is an Lt-term, they are therefore elementary equivalent over vK in the language L.

We will now show that Lv ≡Kv F w. Since Av = Aw = Fp and since Kv is an algebraic extension of L Av, any finite subextension of Kv/Av is definable over Fp, using the language . Since Lv ≡Fp F w, we

thus get that Lv ≡Kv F w. The above arguments show that we are in the situation of Definition 2.4.7 and by Theorem 2.4.8, we

get that (L, v) ≡(K,v) (F, w) in the language L. In particular, (L, v) ≡t (F, w) since t ∈ K, and we are done.

We will now show that Lemma 3.1.2 implies an AKE-principle in Lt relative to the algebraic part.

For this, we will consider monic polynomials f(X) over A≥0, the valuation ring of A. For each such polynomial, let φf be the Lt-sentence defined as

∃X f(X)=0 ∧ v(X) ≥ 0
.

Define

S = {φf | f(X) ∈ A≥0[X], (L, v) |= φf }.

We then have the following.

Theorem 3.1.3. Let (F, w) be a tame field containing A such that (F, w) |= S and such that vL ≡v(t) wF and Lv ≡ F w. Then (L, v) ≡t (F, w).

Proof. Let (F, w) be as described. We want to show that (K, v) is isomorphic as a valued field over A to a subfield of F . By Remark 3.1.1, we have that (A, v) and (A, w) are isomorphic as valued fields. Since both (L, v) and (F, w) are henselian, this isomorphism can be extended to a valued fields isomorphism over A of the henselisations of A in L and F respectively. We now claim that the relative algebraic closure of A in L is isomorphic over A as a field to a subfield of F . Let L′ ⊂ L and F ′ ⊂ F be direct limits of finite extensions of A in L and F respectively, given by irreducible polynomials occurring in S. Since L and F are models of S, we have that L′ and F ′ are isomorphic as fields. Since any finite extension of A is ′ generated by an element which is integral over A≥0, we get that L is the relative algebraic closures of A in L. Since the valuation on a henselian field extends uniquely to algebraic extensions, this isomorphism also preserves the valuation. Hence, we are in the situation of Lemma 3.1.2, and we can conclude the statement.

14 Remark 3.1.4. Note that by Theorem 3.1.3, the copy of K in F will indeed be the relative algebraic closure of A. One could get this immediately by imposing that F |= ¬φf for all monic f ∈ A≥0[X] such

that L |= ¬φf . The reason why we don’t need to do this is because of relative subcompleteness; the fact that (K, v) is a subfield of F is enough to guarantee that it cannot have any proper algebraic extensions in F . The field L in Example 2.4.5 can also be used to show that the assumption of vL/vK being torsion free is necessary. Example 3.1.5. With the notation of Example 2.4.5 and p > 3, let F be an algebraically maximal 1/2 1/3 1/p∞ immediate extension of Fp(t, (st) ,s ) , so F is a tame field with value group equal to vL. Since 1/p∞ 1/3 1/p∞ Lv = Fp(s) is isomorphic to F v = Fp(s ) , we have that Lv ≡ F v. By construction, the

relative algebraic closure of Fp(t) in L is equal to the relative algebraic closure of Fp(t) in F . However,

(F, v) satisfy the Lt-sentence

∃X∃Y ∃Z(X2 = Yt ∧ v(Y )=0 ∧ Z3 = Y )

which is not satisfied in L. In certain cases, it is not necessary to specify the set S in Theorem 3.1.3. Theorem 3.1.6. Let F be a field of characteristic p without any proper finite extension of degree divisible 1/p∞ by p and let (L, v) be F((t )) with the t-adic valuation. Let (F, w) be a tame field containing Fp(t)

such that vL ≡v(t) wF and Lv ≡ F w. Then (L, v) ≡t (F, w).

Proof. Just as in the proof of Theorem 3.1.3, we have that (L, v) and (F, w) contain copies of Fp(t) that

are isomorphic as valued fields with the valuations v|Fp(t) and w|Fp(t) respectively. This isomorphism 1/p∞ can be extended to the perfect hull Fp(t) of Fp(t). Again, we can extend this isomorphism to the 1/p∞ henselisations of Fp(t) in L and F respectively. This isomorphism can be extended to an isomorphism

of valued fields with residue fields being equal to Kv as follows. Let φf be the L-sentence ∃X(f(X)=

0) ∧ (v(X) = 0) where f(X) ∈ Fp(X) has a root in F. Then (L, v) |= φf , so (F, w) |= φf . Adding witnesses to these formulas give an isomorphism of fields, and since the valuation on a henselian field extends uniquely to algebraic extensions, this isomorphism also preserves the valuation. To summarise,

we now have an injective field homomorphism from an algebraic subextension (E, v) of (L, v)/(Fp(t), v) 1 into (F, w), preserving Fp(t), where Ev = Kv and vE = p∞ Z. We identify the image of (E, v) in (F, w) with (E, v) itself.

By Lemma 2.3.3, the relative algebraic closure of Fp(t) in L and F are algebraically maximal Kaplan- sky fields. Hence, (E, v) is also a Kaplansky field and (K, v) is the unique maximal immediate extension of E. Thus, the isomorphism of the copies of E in L and F extends to an isomorphism valued fields between K and a subfield of F and we are in the situation of Lemma 3.1.2, noting that our particular choice of L satisfies the general assumptions of this section.

∞ 3.2 Decidability of F((t1/p ))

We now turn to the question of decidability in Lt for Hahn fields of characteristic p with value group 1 p∞ Z. In this case, we can use the theory of finite automata established in Section 2.5 to show that there is a recursive procedure to determine the set S in Theorem 3.1.3. This approach was suggested by Ehud Hrushovski. We will later see how decidability of general tame Hahn fields of characteristic p can be reduced to this case.

∞ Theorem 3.2.1. Let F be a decidable perfect field of characteristic p. Then the Hahn field F((t1/p )) is decidable in the language Lt.

15 1/p∞ Proof. By Theorem 3.1.3, the Lt-theory of F((t )) is given by the Lval-axioms for tame fields of L 1 L characteristic p, the og(v(t))-theory of p∞ Z and the set S of one variable positive existential t-sentences 1/p∞ L L 1 satisfied by F((t )). Since the ring-theory of F and the og(v(t))-theory of p∞ Z are decidable, so 1/p∞ is the Lval-theory of Fp((t )). In particular, they are recursively enumerable. Hence, it is enough to 1/p∞ show that the set S is recursively enumerable to conclude that the Lt-theory of F((t )) is recursively enumerable. In other words, it is enough to show that there is an algorithm which takes any polynomial f(X) ∈ 1/p∞ Fp[t] as input and returns TRUE if f has a root in F[[t ]] and FALSE otherwise. This algorithm is constructed as follows.

∞ 1. Use Theorem 3.1.6 to decide how many unique roots f has in F¯[[t1/p ]]. Call this number m. By 1/p∞ 1/p∞ Lemma 2.1.1, any root of f in F¯[[t ]] is contained in F¯p((t )). Hence, all such roots lie in 1/p∞ deg(f)! Fq[[t ]], where q = p . Let R be an empty list, to which we will add the roots of f in the subsequent steps.

2. Go through well-ordered DFAOs with output alphabet Fq and check if they are roots to f. By Remark 2.5.21 and by the fact that arithmetic of DFAO as described in Section 2.5 is effective, this is an effective procedure. When a root M is found, proceed to the next step.

3. Check if M is different from all roots in R by computing the DFAO M − N for all N ∈ R. If M − N is nonzero for all N ∈ R, add M to R.

4. If |R| = m, proceed to the next step. Otherwise, repeat step two. Note that |R| will eventually 1/p∞ be equal to m since all roots in Fq[[t ]] are represented by DFAOs, hence this process will eventually proceed to the next step.

∞ 5. Check if any DFAO in R represents a root in F[[t1/p ]]. For a given DFAO M in R with output

function τ and reachable states A, this amounts to checking τ(A) ∩ Fq is a subset of F. As noted

in Remark 2.5.15, the set A is computable from M. Since elements in Fq are definable over Fp in the language of rings and since F is decidable by assumption, we get that there is a decission

procedure to determine if τ(A) ∩ Fq is a subset of F or not.

∞ We conclude that S is recursively enumerable, so F((t1/p )) is decidable as claimed.

3.3 A result by Rayner

Γ Definition 3.3.1. Let x ∈ F((t )). Let r ∈ Γ be in the support of x. Let S be the support of x

1/p∞ As we have seen there are elements in the relative algebraic closure of Fp(t) in Fp((t )) that require infinite ramification for p, for example the roots of Xp − X − t−1. It was established by Rayner that this is not the case for primes different than p [Ray68]. We will give a new proof of this result using additive polynomials. This method will also give an effective bound for the ramification away from p, which will be necessary for our purposes. n pi F Let P (X)= Pi=0 aiX ∈ p(t)[X] be an additive polynomial and write

IP := {i ∈{0,...,n} | ai 6=0}.

i For i ∈ IP , we denote by γi the function on R ∪ {∞} sending r to p r + v(ai).

16 Definition 3.3.2. We say that r ∈ R ∪ {∞} is a point of intersection of P if there are distinct i

and j in IP such that

γi(r)= γj (r) = min{γk(r)}. k

Theorem 3.3.3. Let F be a field of characteristic p and let P (X) ∈ Fp(t)[X] be an additive polynomial. Let q be a prime different from p. If x ∈ F((tQ)) is a root of P and r ∈ Q is such that q ramifies at r in x, then r is a point of intersection of P .

n pi Proof. Write P = Pi=0 aiX . Let r ∈ Γ be in the support of x and suppose that q 6= p is a prime which γ1 γ2 ramifies in x at r. Let c1t be the initial term of P (x

P (x)= P (x

γ1 γ2 r we see that c1t + c2t =0. In particular, γ1 = γ2. Let ζrt be the initial term of x − x

min{γi(r)} = γℓ(r). i∈IP Let J = i ∈ IP γi(r)= γℓ(r) .

We than have that one of the following hold.

k r p i γ2 X ai(ζrt ) = c2t (1) i∈J r pki X ai(ζrt ) =0. (2) i∈J

Suppose that (1) holds. Then, we have γ − v(a ) r = 2 i , pki

contradicting the assumption on q. Hence, we can assume (2) holds. If |J| = 1 we get that ζr = 0, contradicting r being in the support of x. Hence, we get that |J| > 1. By definition, this means that r is a point of intersection of P .

1 Let Γm = mp∞ Z.

Corollary 3.3.4. Let f(X) ∈ Fp(t)[X] and let F be a field. Then, there is a natural number m such that any root of f in F((tQ)) is already in F((tΓm )). Furthermore, m is computable from f.

Proof. Let deg(f)= n. By Ore’s lemma, there is a computable additive polynomial

n pki P (X)= X aiX ∈ Fp(t)[X] i=0 such that f(X) divides P (X). Suppose that x ∈ F((tQ)) is a root of f(X), so in particular a root of

e1 eℓ P (X). Let S be the finite points of intersection of P . Let q1 ,...,qℓ be the prime powers coprime to p occurring as factors in the denominators of the reduced fractions of elements in S. By Theorem 3.3.3 and by definition of a prime ramifying in x, any r = a/b on reduced form in the support of x is such

ℓ ei that the prime power factors of b coprime to p divides m = Qi=1 qi . In other words, the support of x is contained in Γm, and we are done.

∞ F¯ Γm Remark 3.3.5. Rayner shows the existence of the bound in Corollary 3.3.4 by showing that Sm=1 ((t )) is closed under Artin-Schreier extensions. An alternative proof was given by Poonen in [Poo93, Corollary

17 7], which uses the following argument. Let S be the set of automorphisms on F((tQ)) given by send- γ γ Q Z F ing Pγ aγt to Pγ ζ(γ)aγ t , where ζ is a homomorphism from / to the roots of unity in . Then ∞ F¯ Γm F Q Sm=1 ((t )) is the subfield of ((t )) consisting of elements with finite orbits under the action of S. This is an algebraically closed field by [Poo93, Lemma 5]. It was noted by Konstantinos Kartas that effectiveness also follows from the proof by Poonen, since high ramification away from p gives too many roots of f. While this approach would be enough for our main decidability results, using Theorem 3.3.3 gives knowledge of where in the support ramification can occur, which might be of computaional interest for further research.

3.4 Decidability of general positive characteristic tame Hahn fields

We now turn to the question of decidability when there is ramification at primes different from the characteristic. Throughout this section, we will write for any m ∈ N,

1 Γ = Z. m mp∞

We start with the following observation.

Corollary 3.4.1. Let F be a decidable perfect field of characteristic p. Then, for any m ∈ N, we have that F((tΓm )) is decidable.

Proof. This follows from Theorem 3.2.1, with t1/m in place of t.

Theorem 3.4.2. Let (L, v) be a tame field containing Fp(t). Suppose that Lv and vL are decidable in Lring and Log(v(t) respectively. Let F be a perfect decidable subfield of Lv containing the relative Γ algebraic closure of Fp in Lv and let Γ be the relative divisible hull of hv(t)i in vL. Suppose that F((t )) is a subfield of (L, v). Then, (L, v) is decidable in Lt

Proof. Again, it is enough to show that there is a decision procedure for the set S defined before Theorem Γ 3.1.3. To this end, we first note that the relative algebraic closure of Fp(t) in K is contained in F((t )). Indeed, this follows by the assumptions on F and Γ and the fact that Hahn fields are algebraically maximal.

Now, let f(X) ∈ Fp[t][X] be monic. By Corollary 3.3.4, there is a computable bound m such that

Q Γm 1 any root of f in F[[t ]] is already in F[[t ]], where Γm = mp∞ Z as above. Let U be the set of prime power factors of m. Let V be the set of prime powers qe with q 6= p such that v(t)/qe ∈ vK. Then V is decidable by decidability of vK. Since U is computable and V is decidable, we get that

e1 eℓ U ∩ V = {q1 ,...,qℓ }

is computable. Define ℓ ′ ei m = Y qi . i=1

Since F[[tΓ ]] ⊂ F[[tQ]], any root of f in F[[tΓ ]] is already in F[[tΓm ]]. By construction, we have that F[[tΓm ]] ∩ F[[tΓ ]] = F[[tΓm′ ]]. We can thus use the decision procedure from Corollary 3.4.1 for F((tΓm′ )) to determine if f has a root in F[[tΓ ]] or not. Since any root x ∈ L of f is already in F((tΓ )) as noted above, we can now use this same decision procedure to determine if f has a root in K or not, and we are done.

We now get Theorem 1 as an immediate consequence of Theorem 3.4.2, since F((tG)) is a subfield of F((tΓ )), with G being the relative divisible hull of hv(t)i in Γ .

18 Acknowledgment

I wish to thank Jochen Koenigsmann for introducing me to this research question and for continued guidance and proofreading throughout the work. I also wish to thank Ehud Hrushovski for insightful comments, in particular for suggesting to use the work on finite automata. Finally, I wish to thank Konstantinos Kartas for proofreading and many fruitful discussions.

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Mathematical Institute, Woodstock Road, Oxford OX2 6GG. E-mail address: [email protected]

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