9pp, to appear in JSL. Several corrections in new version; Remark (ii) replaced by Example 11 and Remark 12. Thanks to referee for two very useful reports
We study the automorphism group of the algebraic closure of a substructure A of a pseudo-finite field F. We show that the behavior of this group, even when A is large, depends essentially on the roots of unity in F. For almost all completions of the theory of pseudo-finite fields we show that algebraic closure agrees with definable closure, as soon as A contains the relative algebraic closure of the prime field.
We give a criterion when a polynomial $x^n-g$ is irreducible over a pseudofinite field. As an application we give an explicit description of algebraic closure of some pseudofinite fields of zero characteristic.
We explore a notion of pseudofinite dimension, introduced by Hrushovski and Wagner, on an infinite ultraproduct of finite structures. Certain conditions on pseudofinite dimension are identified that guarantee simplicity or supersimplicity of the underlying theory, and that a drop in pseudofinite dimension is equivalent to forking. Under a suitable assumption, a measure-theoretic condition is shown to be equivalent to local stability. Many examples are explored, including vector spaces over finite fields viewed as 2-sorted finite structures, and homocyclic groups. Connections are made to produc…
We show that in an ultraproduct of finite fields, the mod-$n$ nonstandard size of definable sets varies definably in families. Moreover, if $K$ is any pseudofinite field, then one can assign "nonstandard sizes mod $n$" to definable sets in $K$. As $n$ varies, these nonstandard sizes assemble into a definable strong Euler characteristic on $K$, taking values in the profinite completion $\hat{\mathbb{Z}}$ of the integers. The strong Euler characteristic is not canonical, but depends on the choice of a nonstandard Frobenius. When $\operatorname{Abs}(K)$ is finite, the Euler characteristic has som…
In constructive algebra one cannot in general decide the irreducibility of a polynomial over a field K. This poses some problems to showing the existence of the algebraic closure of K. We give a possible constructive interpretation of the existence of the algebraic closure of a field in characteristic 0 by building, in a constructive metatheory, a suitable site model where there is such an algebraic closure. One can then extract computational content from this model. We give examples of computation based on this model.
We apply model theoretic methods to the problem of existence of countable universal graphs with finitely many forbidden connected subgraphs. We show that to a large extent the question reduces to one of local finiteness of an associated''algebraic closure'' operator. The main applications are new examples of universal graphs with forbidden subgraphs and simplified treatments of some previously known cases.
We study algebraic closure and its relation with definable closure in free groups and more generally in torsion-free hyperbolic groups. Given a torsion-free hyperbolic group G and a nonabelian subgroup A of G, we describe G as a constructible group from the algebraic closure of A along cyclic subgroups. In particular, it follows that the algebraic closure of A is finitely generated, quasiconvex and hyperbolic. Suppose that G is free. Then the definable closure of A is a free factor of the algebraic closure of A and the rank of these groups is bounded by that of G. We prove that the algebraic c…
We investigate the computability of algebraic closure and definable closure with respect to a collection of formulas. We show that for a computable collection of formulas of quantifier rank at most $n$, in any given computable structure, both algebraic and definable closure with respect to that collection are $Σ^0_{n+2}$ sets. We further show that these bounds are tight.
We show that the generic automorphism is axiomatisable in the green field of Poizat (once Morleyised) as well as in the bad fields which are obtained by collapsing this green field to finite Morley rank. As a corollary, we obtain "bad pseudofinite fields" in characteristic 0. In both cases, we give geometric axioms. In fact, a general framework is presented allowing this kind of axiomatisation. We deduce from various constructibility results for algebraic varieties in characteristic 0 that the green and bad fields fall into this framework. Finally, we give similar results for other theories ob…
We prove that the existential theory of any function field $K$ of characteristic $p> 0$ is undecidable in the language of rings provided that the constant field does not contain the algebraic closure of a finite field. We also extend the undecidability proof for function fields of higher transcendence degree to characteristic 2 and show that the first-order theory of {\bf any} function field of positive characteristic is undecidable in the language of rings without parameters.
We prove some unconditional cases of the Existential Closedness problem for the modular $j$-function. For this, we show that for any finitely generated field we can find a "convenient" set of generators. This is done by showing that in any field equipped with functions replicating the algebraic behaviour of the modular $j$-function and its derivatives, one can define a natural closure operator in three equivalent different ways.
Einstein Institute of Mathematics of the Hebrew University of Jerusalem（耶路撒冷希伯来大学爱因斯坦数学研究所）；Department of Mathematics “Giuseppe Peano”, University of Torino（都灵大学“朱塞佩·皮亚诺”数学系）
We use a variation on Mason's $α$-function as a pre-dimension function to construct a not one-based $ω$-stable plane $P$ (i.e. a simple rank $3$ matroid) which does not admit an algebraic representation (in the sense of matroid theory) over any field. Furthermore, we characterize forking in $Th(P)$, we prove that algebraic closure and intrinsic closure coincide in $Th(P)$, and we show that $Th(P)$ fails weak elimination of imaginaries, and has Morley rank $ω$.
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.
We prove that a semialgebraically connected affine Nash group over a real closed field R is Nash isogenous to the semialgebraically connected component of the group H(R) of R-points of some algebraic group H defined over R. In the case when R is the field of real numbers this result was claimed in the paper "Groups definable in local fields and pseudofinite fields", Israel J. Math. 85 (1994) by the same two authors, but a mistake in the proof was recently found, and the new proof we obtained has the advantage of being valid over an arbitrary real closed field. We also extend the result to not …
This paper analyzes theorems about algebraic field extensions using the techniques of reverse mathematics. In section 2, we show that $\mathsf{WKL}_0$ is equivalent to the ability to extend $F$-automorphisms of field extensions to automorphisms of $\bar{F}$, the algebraic closure of $F$. Section 3 explores finitary conditions for embeddability. Normal and Galois extensions are discussed in section 4, and the Galois correspondence theorems for infinite field extensions are treated in section 5.
These notes form part of a joint research project on the logic of fields with many valuations, connected by a product formula. We define such structures and name them {\em globally valued fields} (GVFs). This text aims primarily at a proof that {\em the canonical GVF structure on $k(t)^{alg}$ is existentially closed}. This can be read as saying that a variety {\em with a distinguished curve class} is a good approximation for a formula in the language of GVFs, in the same way that a variety is close to a formula for the theory ACF of algebraically closed fields.
Pseudoexponential fields are exponential fields similar to complex exponentiation satisfying the Schanuel Property, which is the abstract statement of Schanuel's Conjecture, and an adapted form of existential closure. Here we show that if we remove the Schanuel Property and just care about existential closure, it is possible to create several existentially closed exponential functions on the algebraic numbers that still have similarities with complex exponentiation. The main difficulties are related to the arithmetic of algebraic numbers, and they can be overcome with known results about speci…
Let $k$ be a field of positive characteristic and $K = k(V)$ a function field of a variety $V$ over $k$ and let ${\mathbf A}_K$ be a ring of adéles of $K$ with respect to a cofinite set of the places on $K$ corresponding to the divisors on $V$. Given a Drinfeld module $Φ:{\mathbb F}[t] \to \operatorname{End}_K({\mathbb G}_a)$ over $K$ and a positive integer $g$ we regard both $K^g$ and ${\mathbf A}_K^g$ as $Φ({\mathbb F}_p[t])$-modules under the diagonal action induced by $Φ$. For $Γ\subseteq K^g$ a finitely generated $Φ(\F_p[t])$-submodule and an affine subvariety $X \subseteq \bG_a^g$ define…
Suppose $G$ is a finite group and $A\subseteq G$ is such that $\{gA:g\in G\}$ has VC-dimension strictly less than $k$. We find algebraically well-structured sets in $G$ which, up to a chosen $ε>0$, describe the structure of $A$ and behave regularly with respect to translates of $A$. For the subclass of groups with uniformly fixed finite exponent $r$, these algebraic objects are normal subgroups with index bounded in terms of $k$, $r$, and $ε$. For arbitrary groups, we use Bohr neighborhoods of bounded rank and width inside normal subgroups of bounded index. Our proofs are largely model theoret…
This paper is the second in a series of three, the aim of which is to construct algebraic geometry over a free metabelian Lie algebra $F$. For the universal closure of free metabelian Lie algebra of finite rank $r \ge 2$ over a finite field $k$ we find a convenient set of axioms in the language of Lie algebras $L$ and the language $L_{F}$ enriched by constants from $F$. We give a description of: * The structure of finitely generated algebras from the universal closure of $F_r$ in both $L$ and $L_{F_r}$ * The structure of irreducible algebraic sets over $F_r $ and respective coordinate algebras…
I give an algebraic proof that the exponential algebraic closure operator in an exponential field is always a pregeometry, and show that its dimension function satisfies a weak Schanuel property. A corollary is that there are at most countably many essential counterexamples to Schanuel's conjecture.
