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We prove that every ultraproduct of $p$-adics is inp-minimal (i.e., of burden $1$). More generally, we prove an Ax-Kochen type result on preservation of inp-minimality for Henselian valued fields of equicharacteristic $0$ in the RV language., Comment: v.2: 15 pages, minor corrections and presentation improvements; accepted to the Journal of Symbolic Logic

In this note we give sentences φK in the language of fields which describe the isomorphy type of K among finitely generated fields, provided the Kronecker dimension of K is ≤ 2, or equivaelntly, K is the function field of a curve over a global field, thus extending the corresponding results by Rumely concerning global fields. This closes the gap from Scanlon’s [Sc] approach to proving what he calls Pop’s Conjecture for K as above.

We study the existential (and parts of the universal-existential) theory of equicharacteristic henselian valued fields. We prove, among other things, an existential Ax-Kochen-Ershov principle, which roughly says that the existential theory of an equicharacteristic henselian valued field (of arbitrary characteristic) is determined by the existential theory of the residue field; in particular, it is independent of the value group. As an immediate corollary, we get an unconditional proof of the decidability of the existential theory of $\mathbb{F}_{q}((t))$.

We give a construction of quasiminimal fields equipped with pseudo-analytic maps, generalising Zilber's pseudo-exponential function. In particular we construct pseudo-exponential maps of simple abelian varieties, including pseudo-$\wp$-functions for elliptic curves. We show that the complex field with the corresponding analytic function is isomorphic to the pseudo-analytic version if and only the appropriate version of Schanuel's conjecture is true and the corresponding version of the strong exponential-algebraic closedness property holds. Moreover, we relativize the construction to build a model over a fairly arbitrary countable subfield and deduce that the complex exponential field is quasiminimal if it is exponentially-algebraically closed. This property asks only that the graph of exponentiation have non-trivial intersection with certain algebraic varieties but does not require genericity of these points. Furthermore Schanuel's conjecture is not required as a condition for quasiminimality., v3: Substantial improvements to the organisation and presentation

Denef and Loeser defined a map from the Grothendieck ring of sets definable in pseudo-finite fields to the Grothendieck ring of Chow motives, thus enabling to apply any cohomological invariant to these sets. We generalize this to perfect, pseudo algebraically closed fields with pro-cyclic Galois group. In addition, we define some maps between different Grothendieck rings of definable sets which provide additional information, not contained in the associated motive. In particular we infer that the map of Denef-Loeser is not injective., Comment: 15 pages, to appear in JSL; minor corrections and one proof replaced by a sketch of proof and a reference

We give a complete description of minimal groups infinitely definable in separably closed fields of finite degree of imperfection. In particular we answer positively the question of the existence of such a group with infinite transcendence degree (i.e., a minimal group with non thin generic).

I prove a version of Schanuel's conjecture for Weierstrass equations in differential fields, answering a question of Zilber, and show that the linear independence condition in the statement cannot be relaxed.

We answer a question of Pillay and Ziegler and construct a type of $U$-rank 1 (of the theory of separably closed fields) which is thin but not very thin.

We relate the integrability of vector fields, and of the vanishing of $p$-torsion, to model-theoretic questions concerning separably closed fields, endowed canonically with a derivation. While each differential field $(F_p(t)^s,D_p)$ is known to be decidable, we show that the asymptotic theory of these fields as a class is undecidable in a strong sense. This precludes a geometric answer to certain generalizations of the Grothendieck-Katz conjecture.

The study of pseudo-algebraically closed fields (henceforth called PAC) started with the work of J. Ax on finite and pseudo-finite fields [1]. He showed that the infinite models of the theory of finite fields are exactly the perfect PAC fields with absolute Galois group isomorphic to , and gave elementary invariants for their first order theory, thereby proving the decidability of the theory of finite fields. Ax's results were then extended to a larger class of PAC fields by M. Jarden and U. Kiehne [21], and Jarden [19]. The final word on theories of PAC fields was given by G. Cherlin, L. van den Dries and A. Macintyre [10], see also results by Ju. Ershov [13], [14]. Let K be a PAC field. Then the elementary theory of K is entirely determined by the following data:• The isomorphism type of the field of absolute numbers of K (the subfield of K of elements algebraic over the prime field).• The degree of imperfection of K.• The first-order theory, in a suitable ω-sorted language, of the inverse system of Galois groups al(L/K) where L runs over all finite Galois extensions of K.They also showed that the theory of PAC fields is undecidable, by showing that any graph can be encoded in the absolute Galois group of some PAC field. It turns out that the absolute Galois group controls much of the behaviour of the PAC fields. I will give below some examples illustrating this phenomenon.

We show that the class of separably closed fields with a generic automorphism is an elementary class, whose theory is model complete in a natural extension of the language of fields with an automorphism. We describe the completions of this theory and obtain some results on types, imaginaries, and modularity.

In this paper, we look at strongly minimal sets definable in a differentially closed field of characteristic 0. In [3], Hrushovski and Sokolović show that such sets are essentially Zariski geometries. Thus either thre is a definable strongly minimal field nonorthogonal to $D$, or $D$ is locally modular and nontrivial, or $D$ is trivial. We show that the strongly minimal sets defined by a certain family of differential equations are trivial. We also prove a theorem wich provides a test for the orthogonality of types over an ordinary differential field.

We transpose Delon’s analysis of types in valued fields to unramified henselian valued fields of mixed characteristic, by using coefficient maps of order n. This yields the Ax-Kochen-Ershov transfer principle for the independence property in this class of valued fields.