Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.40251

PAC fields and difference fields with generic automorphisms

Özlem Beyarslan·Piotr Kowalski
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the existence of model companions after adjoining an automorphism to field theories in two complementary settings. For bounded PAC fields of characteristic zero containing all roots of unity and admitting a cyclic Galois quotient of prime order, a suitable choice of countably many constants gives a model-complete field theory which has no model companion after adding an automorphism fixing the named constants. This applies, in particular, when the absolute Galois group is free profinite or free pro-$p$ of finite positive rank. Secondly, for every completion of $\ACFA$ in every characteristic, adjoining a second commuting automorphism has no model companion. The two explicit obstructions use a common ordered product of inertia automorphisms along an infinite orbit, combined respectively with an equivariant PAC completion and an algebraically closed difference-field embedding.

Explore related subjects

math.LO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Özlem Beyarslan, Piotr Kowalski. 2026-09-30. PAC fields and difference fields with generic automorphisms. https://arxiv.org/abs/2609.40251

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Higher amalgamation in $\mathrm{ACFA}^{+}$

We show two results on higher amalgamation in the theory $\mathrm{ACFA}^{+}$, the model companion of the theory of difference fields with an additive character (added as a continuous logic predicate) on the fixed field in characteristic 0. On one hand, we show that the non-trivial condition for 3-amalgamation established in a preceding paper is not sufficient for 4-amalgamation. On the other hand, we show that when working over substructures whose $\mathcal{L}_σ$-reduct is a model of $\mathrm{ACFA}$, $n$-amalgamation holds for all $n\geq 3$.

math.LO↗
arXiv2026

Some results on NIP groups and their Ellis groups

This paper has several parts. We begin by developing a theory of `piecewise (strong) f-genericity' in NIP groups, where we call a definable set piecewise (strong) f-generic if some union of finitely many translates of it is (strong) f-generic. We show that, in an NIP group, the definable sets that are not piecewise (strong) f-generic form an ideal. Our hope is that the corresponding piecewise (strong) f-generic types can provide a substitute in arbitrary NIP groups for the (strong) f-generic types of definably amenable NIP groups, and in the rest of the paper we give several applications. Two of the applications deal with the Ellis group of an NIP group. Let $T$ be an NIP theory, $G$ a definable group, and $M$ a model. In our first result we show that the size of the Ellis group of $G(M)$ is bounded above by $2^{|T|}$, independent of the choice of $M$, giving a substantial step towards the question of whether the isomorphism type is independent of $M$. In our second result, inspired by a theorem of Hrushovski, we show that, if $T$ and $M$ are countable and the formulas of $T$ have uniformly bounded VC-codensity, then the Ellis group of $G(M)$ has `finite Archimedean rank', ie its connected component is profinite-by-Lie. A crucial tool for us in both results is the recent result of Chernikov-Gannon-Krupiński and Basso-Zucker that the $τ$-topology on the Ellis group is Hausdorff. Finally, we use our techniques to obtain a `local' result valid in arbitrary NIP theories, without the assumption of uniformly bounded VC-codensity: for any `bi-invariant' formula $ϕ(x,y)$, the group $G/G^{00}_ϕ$ has finite Archimedean rank. More precisely, if the VC-codensity of $ϕ(x,y)$ is at most $δ$, then $G/G^{00}_ϕ$ is an inverse limit of compact Lie groups of dimension at most $(4δ)^2$. This connects to, though is different than, a question of Hrushovski.

math.LO↗
arXiv2026

Strong failures of club guessing at the successor of a regular cardinal

Starting from a model of $\mathrm{ZFC}$, we force the simultaneous failure of the Very Weak Club Guessing and the $\mho$ principles at $S_κ^{κ^+}$, for any regular cardinal $κ$. Our results are obtained by means of a forcing iteration technique, due to Krueger, that incorporates models as side conditions. At $ω_1$, the failure of these principles is a well-known consequence of $\mathrm{PFA}$.

math.LO↗