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We study when the property that a field is dense in its real and p-adic closures is elementary in the language of rings and deduce that all models of the theory of algebraic fields have this property., Comment: 19 pages

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

We correct a mistake in a lemma in the paper cited in the title and show that it did not affect any of the other results of the paper. To this end, we prove results on linearly disjoint field extensions that do not seem to be commonly known. We give an example to show that a separability assumption in one of these results cannot be dropped (doing so had led to the mistake). Further, we discuss recent generalizations of the original classification of defect extensions.

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 $\mathbb{F}_{q}((t))$.

†† Abstract. Weconstructafibereddimensionfunctioninsometopologicaldierentialfields. §1. Introduction. LetKbeatopologicalL-fieldasdefinedin(6)(seealsosection 3). WearegoingtoconsidertwokindsofexpansionshK,Di,whereDisaderivation onK: onewheretheoperatorDhasapriorinointeractionswiththetopologyon KandtheotheronewhereDinducesacontinuousmapfromKtoK. In thefirst case, we will work in the setting we developed in (6). AssumeK is amodelofauniversalL-theoryTwhichhasamodelcompletionTc. Undercertain hypothesis onTc, weshowedthattheexpansionofT to theL ∪ {D}-theoryTD consistingofT togetherwiththeaxiomsexpressingthatDisaderivation,admits amodel-completionT ∗ c,D whichweaxiomatized. TothetheoryTD ∪Tc,weadded a scheme of axioms (DL), which expressess that each dierent ial polynomial has a zero close to a zero of its associated algebraic polynomial. This scheme (DL) is relatedtotheaxiomscheme(UC)introducedbyM.Tresslintheframeworkof largefields,andalsototheaxiomatization,duetoM.Singer,ofthetheoryofclosed ordereddierentia lfields(CODF). WefurtherobservedthatwheneverTchasthenon-independenceproperty(NIP), thenT ∗ c,D hasNIP.Thisimpliesforinstancethatthedefinablesubsetsinamodelof T ∗ c,D haveaVC-dimension. Here, using results of L. van den Dries, we will show the existence of afibered

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

Let $p$ be a prime. In this paper we give a proof of the following result: A valued field $(K,v)$ of characteristic $p > 0$ is $p$-henselian if and only if every element of strictly positive valuation is of the form $x^p - x$ for some $x \in K$.

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

The main purpose of this article is to demonstrate three techniques for proving algebraicity statements about circle packings. We give proofs of three related theorems: (1) that every finite simple planar graph is the contact graph of a circle packing on the Riemann sphere, equivalently in the complex plane, all of whose tangency points, centers, and radii are algebraic, (2) that every flat conformal torus which admits a circle packing whose contact graph triangulates the torus has algebraic modulus, and (3) that if R is a compact Riemann surface of genus at least 2, having constant curvature -1, which admits a circle packing whose contact graph triangulates R, then R is isomorphic to the quotient of the hyperbolic plane by a subgroup of PSL_2(real algebraic numbers). The statement (1) is original, while (2) and (3) have been previously proved in the Ph.D. thesis of McCaughan. Our first proof technique is to apply Tarski's Theorem, a result from model theory, which says that if an elementary statement in the theory of real-closed fields is true over one real-closed field, then it is true over any real closed field. This technique works to prove (1) and (2). Our second proof technique is via an algebraicity result of Thurston on finite co-volume discrete subgroups of the orientation-preserving-isometry group of hyperbolic 3-space. This technique works to prove (1). Our first and second techniques had not previously been applied in this area. Our third and final technique is via a lemma in real algebraic geometry, and was previously used by McCaughan to prove (2) and (3). We show that in fact it may be used to prove (1) as well., 16 pages, 1 figure

Fields of characteristic zero with several commuting derivations can be treated as fields equipped with a space of derivations that is closed under the Lie bracket. The existentially closed instances of such structures can then be given a coordinate-free characterization in terms of differential forms. The main tool for doing this is a generalization of the Frobenius Theorem of differential geometry.

Given any fieldK, there is a function fieldF/Kin one variable containing definable transcendental overK, i.e., elements inF / Kfirst-order definable in the language of fields with parameters fromK. Hence, the model-theoretic and the field-theoretic relative algebraic closure ofKinF do notcoincide. E.g., ifKis finite, the model-theoretic algebraic closure ofKin the rational function fieldK(t)isK(t).For the proof, diophantine ∅-definability ofKinFis established for any function fieldF/Kin one variable, providedKis large, orK×/(K×)nis finite for some integern> 1 coprime tochar K.

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).

A theory $T$ is shown to have an ICT pattern of depth $k$ in variables $\xbar$ iff it interprets some $k$-maximum VC class in $|\xbar|$ parameters., Comment: 14 pages. This is a revised and extended version of "A note on dp-rank"

We compare several definitions of the Galois group of a linear difference equation that have arisen in algebra, analysis and model theory and show, that these groups are isomorphic over suitable fields. In addition, we study properties of Picard-Vessiot extensions over fields with not necessarily algebraically closed subfields of constants.

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 study the universal cover of the complex one-dimensional torus as a model-theoretic structure in a natural language. We consider also abstract covers of one-dimensional tori over algebraically closed fields of characteristic zero. The main result states that the structures can be described uniquely up to isomorphism by a simple (non-first order) sentence, given a fixed uncountable cardinality of the underlying field. The proof reduces to the Kummer theory of cyclic Galois extensions over some, not necessarily finitely generated fields., 25 pages, Latex

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.