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We study various universal-existential fragments of first-order theories of fields, in particular of function fields and of equicharacteristic henselian valued fields. For example we discuss to what extent the theory of a field k determines the universal-existential theories of the rational function field over k and of the field of Laurent series over k, and we find various many-one reductions between such fragments., Comment: added Remark 6.18, otherwise minor changes

We discuss the common existential theory of all or almost all completions of a global function field., Comment: minor changes to Introduction and Remark 6.2

Recently, Widmer introduced a new sufficient criterion for the Northcott property on the finiteness of elements of bounded height in infinite algebraic extensions of number fields. We provide a simplification of Widmer's criterion when the extension is abelian, and use this to exhibit fields with the Northcott property that do not satisfy Widmer's new criterion. We also show how the construction of pseudo algebraically closed fields with the Northcott property, carried out in previous work using the first version of Widmer's criterion, can be simplified., Comment: Minor changes. To appear in Proc. AMS

We set up general machinery to study interpretations of fragments of theories. We then apply this to existential fragments of theories of fields, and especially of henselian valued fields. As an application we prove many-one reductions between various existential theories of fields. In particular we exhibit several theories of fields many-one equivalent to the existential theory of $\mathbb{Q}$., Comment: Results unchanged. Added explanations, in particular in Section 2. Slightly rearranged Section 3. To appear in Israel Journal of Mathematics

We study the preservation of the Hilbert property and of the weak Hilbert property under base change in field extensions. In particular we show that these properties are preserved if the extension is finitely generated or Galois with finitely generated Galois group, and we also obtain some negative results.

We study rational points on ramified covers of abelian varieties over certain infinite Galois extensions of $\mathbb{Q}$. In particular, we prove that every elliptic curve $E$ over $\mathbb{Q}$ has the weak Hilbert property of Corvaja-Zannier both over the maximal abelian extension $\mathbb{Q}^{\rm ab}$ of $\mathbb{Q}$, and over the field $\mathbb{Q}(A_{\rm tor})$ obtained by adjoining to $\mathbb{Q}$ all torsion points of some abelian variety $A$ over $\mathbb{Q}$., Comment: preliminaries section shortened, minor corrections

We study the existential theory of equicharacteristic henselian valued fields with a distinguished uniformizer. In particular, assuming a weak consequence of resolution of singularities, we obtain an axiomatization of - and therefore an algorithm to decide - the existential theory relative to the existential theory of the residue field. This is both more general and works under weaker resolution hypotheses than the algorithm of Denef and Schoutens, which we also discuss in detail. In fact, the consequence of resolution of singularities our results are conditional on is the weakest under which they hold true., Comment: New Remark 4.18 and expanded Remark 2.4

We study the minimal number of ramified primes in Galois extensions of rational function fields over finite fields with prescribed finite Galois group. In particular, we obtain a general conjecture in analogy with the well studied case of number fields, which we establish for abelian, symmetric and alternating groups in many cases., Comment: v3: minor changes

We study the minimal number of existential quantifiers needed to define a diophantine set over a field and relate this number to the essential dimension of the functor of points associated to such a definition., Comment: Expanded results in Section 7, otherwise minor changes

Bombieri and Zannier gave an effective construction of algebraic numbers of small height inside the maximal Galois extension of the rationals which is totally split at a given finite set of prime numbers. They proved, in particular, an explicit upper bound for the lim inf of the height of elements in such fields. We generalize their result in an effective way to maximal Galois extensions of number fields with given local behaviour at finitely many places.

We show that the set of algebraic extensions $F$ of $\mathbb{Q}$ in which $\mathbb{Z}$ or the ring of integers $\mathcal{O}_F$ are definable is meager in the set of all algebraic extensions., Comment: Added some remarks. Accepted for publication in the Notre Dame Journal of Formal Logic

The first aim of this note is to fill a gap in the literature by proving that, given a global field $K$ and a finite set $\mathcal{S}$ of primes of $K$, every finite split embedding problem $G \rightarrow {\rm{Gal}}(L/K)$ over $K$ with nilpotent kernel has a solution ${\rm{Gal}}(F/K) \rightarrow G$ such that all primes in $\mathcal{S}$ are totally split in $F/L$. We then apply this to inverse Galois theory over division rings. Firstly, given a number field $K$ of level at least $4$, we show that every finite solvable group occurs as a Galois group over the division ring $H_K$ of quaternions with coefficients in $K$. Secondly, given a finite split embedding problem with nilpotent kernel over a finite field $K$, we fully describe for which automorphisms $\sigma$ of $K$ the embedding problem acquires a solution over the skew field of fractions $K(T, \sigma)$ of the twisted polynomial ring $K[T, \sigma]$.

We construct infinite Galois extensions $K$ of $\mathbb{Q}$ that satisfy the Northcott property on elements of small height, and where this property can be deduced solely from the splitting behavior of prime numbers in $K$. We also give examples of Galois extensions of $\mathbb{Q}$ which have finite local degree at all prime numbers and do not satisfy the Northcott property., Comment: strengthened Theorem 1.2, added remarks on undecidability results

Let $A$ be a non-zero abelian variety over a field $F$ that is not algebraic over a finite field. We prove that the rational rank of the abelian group $A(F)$ is infinite when $F$ is large in the sense of Pop (also called ample). The main ingredient is a deduction of the 1-dimensional case of the relative Mordell-Lang conjecture from a result of R\"ossler.

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

Siegel proved that every totally positive element of a number field K is the sum of four squares, so in particular the Pythagoras number is uniformly bounded across number fields. The p-adic Kochen operator provides a p-adic analogue of squaring, and a certain localisation of the ring generated by this operator consists of precisely the totally p-integral elements of K. We use this to formulate and prove a p-adic analogue of Siegel's theorem, by introducing the p-Pythagoras number of a general field, and showing that this number is uniformly bounded across number fields. We also generally study fields with finite p-Pythagoras number and show that the growth of the p-Pythagoras number in finite extensions is bounded.

The classical Artin--Whaples approximation theorem allows to simultaneously approximate finitely many different elements of a field with respect to finitely many pairwise inequivalent absolute values. Several variants and generalizations exist, for example for finitely many (Krull) valuations, where one usually requires that these are independent, i.e. induce different topologies on the field. Ribenboim proved a generalization for finitely many valuations where the condition of independence is relaxed for a natural compatibility condition, and Ershov proved a statement about simultaneously approximating finitely many different elements with respect to finitely many possibly infinite sets of pairwise independent valuations. We prove approximation theorems for infinite sets of valuations and orderings without requiring pairwise independence., Comment: Correction of typographical errors, simplified exposition in the last two sections

A central conjecture in inverse Galois theory, proposed by D\`{e}bes and Deschamps, asserts that every finite split embedding problem over an arbitrary field can be regularly solved. We give an unconditional proof of a consequence of this conjecture, namely that such embedding problems can be regularly solved if one waives the requirement that the solution fields are normal. This extends previous results of M. Fried, Takahashi, Deschamps, and the last two authors concerning the realization of finite groups as automorphism groups of field extensions.

We give three examples of fields concerning the Northcott property on elements of small height: The first one has the Northcott property but its Galois closure does not satisfy the Bogomolov property. The second one has the Northcott property and is pseudo-algebraically closed, i.e. every variety has a dense set of rational points. The third one has bounded local degree at infinitely many rational primes but does not have the Northcott property., Comment: 5 pages

We investigate a function field analogue of a recent conjecture on autocorrelations of sums of two squares by Freiberg, Kurlberg and Rosenzweig, which generalizes an older conjecture by Connors and Keating. In particular, we provide extensive numerical evidence and prove it in the large finite field limit. Our method can also handle correlations of other arithmetic functions and we give applications to (function field analogues of) the average of sums of two squares on shifted primes, and to autocorrelations of higher divisor functions twisted by a quadratic character.

Although the study of the definability of henselian valuations has a long history starting with J. Robinson, most of the results in this area were proven during the last few years. We survey these results which address the definability of concrete henselian valuations, the existence of definable henselian valuations on a given field, and questions of uniformity and quantifier complexity., Comment: Survey article, 8 pages

We give a characterization, in terms of the residue field, of those henselian valuation rings and those henselian valuation ideals that are diophantine. This characterization gives a common generalization of all the positive and negative results on diophantine henselian valuation rings and diophantine valuation ideals in the literature. We also treat questions of uniformity and we apply the results to show that a given field can carry at most one diophantine nontrivial equicharacteristic henselian valuation ring or valuation ideal., Comment: 29 pages

Landau's theorem asserts that the asymptotic density of sums of two squares in the interval $1\leq n\leq x$ is $K/{\sqrt{\log x}}$, where $K$ is the Landau-Ramanujan constant. It is an old problem in number theory whether the asymptotic density remains the same in intervals $|n-x|\leq x^{\epsilon}$ for a fixed $\epsilon$ and $x\to \infty$. This work resolves a function field analogue of this problem, in the limit of a large finite field. More precisely, consider monic $f_0\in \mathbb{F}_q[T]$ of degree $n$ and take $\epsilon$ with $1>\epsilon\geq \frac2n$. Then the asymptotic density of polynomials $f$ in the `interval' ${\rm deg}(f-f_0)\leq \epsilon n$ that are of the form $f=A^2+TB^2$, $A,B\in \mathbb{F}_q[T]$ is $\frac{1}{4^n}\binom{2n}{n}$ as $q\to \infty$. This density agrees with the asymptotic density of such monic $f$'s of degree $n$ as $q\to \infty$, as was shown by the second author, Smilanski, and Wolf. A key point in the proof is the calculation of the Galois group of $f(-T^2)$, where $f$ is a polynomial of degree $n$ with a few variable coefficients: The Galois group is the hyperoctahedral group of order $2^nn!$.

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 discuss definability in the language of rings without parameters of the unique canonical henselian valuation of a field. We show that in most cases where the canonical henselian valuation is definable, it is already definable by a universal-existential or an existential-universal formula., Comment: 17 pages

We discuss definability of henselian valuation rings in the Macintyre language $\mathcal{L}_{\rm Mac}$, the language of rings expanded by n-th power predicates. In particular, we show that henselian valuation rings with finite or Hilbertian residue field are uniformly $\exists$-$\emptyset$-definable in $\mathcal{L}_{\rm Mac}$, and henselian valuation rings with value group $\mathbb{Z}$ are uniformly $\exists\forall$-$\emptyset$-definable in the ring language, but not uniformly $\exists$-$\emptyset$-definable in $\mathcal{L}_{\rm Mac}$. We apply these results to local fields $\mathbb{Q}_p$ and $\mathbb{F}_p((t))$, as well as to higher dimensional local fields.

We define and investigate a family of local-global principles for fields involving both orderings and p-valuations. This family contains the PAC, PRC and PpC fields and exhausts the class of pseudo classically closed fields. We show that the fields satisfying such a local-global principle form an elementary class, admit diophantine definitions of holomorphy domains, and their orderings satisfy the strong approximation property., Comment: final version published in Annals of Pure and Applied Logic, Volume 164, Issue 10, October 2013, Pages 989-1008

We analyze the elementary theory of certain fields of totally S-adic algebraic numbers that were introduced and studied by Geyer-Jarden and Haran-Jarden-Pop. In particular, we provide an axiomatization of these theories and prove their decidability, thereby giving a common generalization of classical decidability results of Jarden-Kiehne, Fried-Haran-V\"olklein and Ershov.

We construct and study fields F with the property that F has infinitely many extensions of some fixed degree, but E*/(E*)^n is finite for every finite extension E of F and every n>0., Comment: 9 pages

Recently, Anscombe and Koenigsmann gave an existential 0-definition of the ring of formal power series F[[t]] in its quotient field in the case where F is finite. We extend their method in several directions to give general definability results for henselian valued fields with finite or pseudo-algebraically closed residue fields., Comment: minor corrections, Theorem 4.1 strengthened, Remark 4.2 added

A variety X over a field K is of Hilbert type if the set of rational points X(K) is not thin. We prove that if f: X\to S is a dominant morphism of K-varieties and both S and all fibers f^{-1}(s), s in S(K), are of Hilbert type, then so is X. We apply this to answer a question of Serre on products of varieties and to generalize a result of Colliot-Th'el`ene and Sansuc on algebraic groups., Comment: We added a corollary on homogeneous spaces, and slightly generalized the formulation of the main result

Let Z{t} be the ring of arithmetic power series that converge on the complex open unit disc. A classical result of Harbater asserts that every finite group occurs as a Galois group over the quotient field of Z{t}. We strengthen this by showing that every finite split embedding problem over Q acquires a solution over this field. More generally, we solve all t-unramified finite split embedding problems over the quotient field of O{t}, where O is the ring of integers of an arbitrary number field K., Comment: 23 pages

Let L be a Galois extension of a countable Hilbertian field K. Although L need not be Hilbertian, we prove that an abundance of large Galois subextensions of L/K are.

We prove a new Hilbertianity criterion for fields in towers whose steps are Galois with Galois group either abelian or a product of finite simple groups. We then apply this criterion to fields arising from Galois representations. In particular we settle a conjecture of Jarden on abelian varieties., Comment: 18 pages, accepted for publication in Journal f\"ur die reine und angewandte Mathematik

Given a finite set $S$ of places of a number field, we prove that the field of totally $S$-adic algebraic numbers is not Hilbertian., Comment: Appendix by Florian Pop is added

Fifteen years after their discovery, ample fields now stand at the center of research in contemporary Galois theory and attract more and more attention also from other areas of mathematics. This survey gives an introduction to the theory of ample fields and discusses open problems.

Pop proved that a smooth curve C over an ample field K that has a K-rational point has |K| many K-rational points. We strengthen this result by showing that there are |K| many K-rational points that do not lie in a given proper subfield, even after applying a rational map. As a consequence we gain insight into the structure of existentially definable subsets of ample fields. In particular, we prove that a perfect ample field has no existentially definable proper infinite subfields., Comment: 8 pages