On n-dependent groups and fields II

TL;DR

This paper advances the understanding of n-dependent groups and fields, providing evidence for the conjecture that all n-dependent fields are dependent, and exploring properties of valued fields, group components, and examples of 2-dependent fields.

Contribution

It proves that infinite n-dependent valued fields of positive characteristic are henselian, generalizes Shelah's Henselianity Conjecture, and introduces new examples of 2-dependent fields with additional structure.

Findings

Infinite n-dependent valued fields of positive characteristic are henselian.

Generalization of Shelah's Henselianity Conjecture in this context.

New examples of 2-dependent fields with additional structure.

Abstract

We continue the study of $n$-dependent groups, fields and related structures, largely motivated by the conjecture that every $n$-dependent field is dependent. We provide evidence towards this conjecture by showing that every infinite $n$-dependent valued field of positive characteristic is henselian, obtaining a variant of Shelah's Henselianity Conjecture in this case and generalizing a recent result of Johnson for dependent fields. Additionally, we prove a result on intersections of type-definable connected components over generic sets of parameters in $n$-dependent groups, generalizing Shelah's absoluteness of $G^{00}$ in dependent theories and relative absoluteness of $G^{00}$ in $2$-dependent theories. In an effort to clarify the scope of this conjecture, we provide new examples of strictly $2$-dependent fields with additional structure, showing that Granger's examples of non-degenerate bilinear forms over dependent fields are $2$-dependent. Along the way, we obtain some purely model-theoretic results of independent interest: we show that $n$-dependence is witnessed by formulas with all but one variable singletons; provide a type-counting criterion for $2$-dependence and use it to deduce $2$-dependence for compositions of dependent relations with arbitrary binary functions (the Composition Lemma); and show that an expansion of a geometric theory $T$ by a generic predicate is dependent if and only if it is $n$-dependent for some $n$, if and only if the algebraic closure in $T$ is disintegrated. An appendix by Martin Bays provides an explicit isomorphism in the Kaplan-Scanlon-Wagner theorem.