Some Asymptotic Results on Fields

Zoé Chatzidakis & Ehud Hrushovski

First set of results. Let Q be the set of all prime powers, and for q = p^{n} ∈ Q, p a prime, let K_{q} be a separably closed, non algebraically closed field of characteristic p > 0, and consider the difference field (K_{q}, σ_{q}), where σ_{q} : x ↦ x^{q}. We consider the theory T consisting of all sentences true in almost all K_{q}, and give an axiomatisation of T. In the language obtained by adding to {+, ·, 0, 1, σ} functions (which will be interpreted as the λ-functions of K if the characteristic is p > 0) and predicates for σ(K)-linear independence, we show that the expansion by definition of T is model-complete. All completions of T are simple, and various results valid in models of ACFA extend.

Second set of results. For each prime p, let F (p) be the differential field (𝔽_{p}(t)^{s}, +, ·, D_{p}), where t is transcendental over 𝔽_{p} and D_{p} is the unique derivation on 𝔽_{p}(t)^{s} satisfying D_{p}t = 1. We let T_{D} be the set of all sentences true in almost all F (p). We show that T_{D} is bi-interpretable with a certain second-order theory of pseudo-finite fields. We also show that T_{D∀} and T_{D∃} are Σ_{2}^{0}-complete.

Addresses Zoé Chatzidakis: Université Paris 7, UFR de mathématiques, Case 7012, 2, place Jussieu, 75251 Paris Cedex 05, France

E-mail: zoe@logique.jussieu.fr

Ehud Hrushovski: Institute of Mathematics, The Hebrew University, Givat Ram, Jerusalem 91904, Israel

E-mail: ehud@math.huji.ac.il