Decidability of positive characteristic tame Hahn fields in Lt Victor Lisinski∗ August 10, 2021 Abstract Γ We show that any positive characteristic tame Hahn field F((t )) containing t is decidable in Lt, the language of valued fields with a constant symbol for t, if F and Γ are decidable. In particular, 1/p∞ Q we obtain decidability of Fp((t )) and Fp((t )) in Lt. This uses a new AKE-principle for equal characteristic tame fields in Lt, building on work by Kuhlmann, together with Kedlaya’s work on finite automata and algebraic extensions of function fields. In the process, we obtain an AKE- principle for tame fields in mixed characteristic and recover a theorem by Rayner on the relative algebraic closure of function fields inside Hahn fields. Contents 1 Introduction 1 2 Preliminaries 2 2.1 Notation and basic algebraic preliminaries . ................. 2 2.2 Modeltheoryanddecidability. ............ 3 2.3 Kaplanskyfields .................................. ....... 4 2.4 Tamefields ....................................... ..... 5 2.5 Finiteautomata .................................. ....... 6 3 Main results 13 3.1 An AKE-principle for tame fields in Lt ............................. 13 ∞ 3.2 Decidability of F((t1/p )) .................................... 15 3.3 AresultbyRayner................................. ....... 16 arXiv:2108.04132v1 [math.LO] 9 Aug 2021 3.4 Decidability of general positive characteristic tame Hahnfields ............... 18 1 Introduction While the first order theory, and in particular decidability, of the p-adic numbers Qp is well understood thanks to the work by Ax and Kochen [AK65] and, independently, Ershov [Ers65], a long standing open problem is that of decidability of the equal characteristic analogue Fp((t)). Some recent progress have been made on this topic when restricting the question to existential decidability. In [DS03], Denef and Shoutens showed that the existential theory of Fp((t)) in the language of rings with a constant symbol for t is decidable assuming resolution of singularities in characteristic p. In [AF16], Anscombe and Fehm ∗The author was funded by an EPSRC award at the University of Oxford, with additional support from the Royal Swedish Academy of Sciences and Corpus Christi College Oxford. 1 showed that the existential theory of Fp((t)) is unconditionally decidable in the language of rings. The results by Anscombe and Fehm uses decidability results on tame fields in the one sorted language of 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<r = X aγ t γ∈Γ<r γ x≤r = X aγt . γ∈Γ≤r γi Similarly, with the notation x = Pi∈I ait , we write x<i = x<ri and x≤i = x≤ri . ∞ 1 Z We write F((t)) in place of F((tZ)) and F((t1/p )) in place of F((t p∞ )). It was noted by Abhyankar F 1/p∞ p −1 −pn that p((t )) admits a root to the Artin-Schreier polynomial 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.