Abstract
AbstractWe continue the study ofn-dependent groups, fields and related structures, largely motivated by the conjecture that everyn-dependent field is dependent. We provide evidence toward this conjecture by showing that every infiniten-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 inn-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 thatn-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 theoryTby a generic predicate is dependent if and only if it isn-dependent for somen, if and only if the algebraic closure inTis disintegrated. An appendix by Martin Bays provides an explicit isomorphism in the Kaplan-Scanlon-Wagner theorem.
Highlights
A classical line of research in model theory, both pure and applied, aims to determine properties of algebraic structures, such as groups and fields, that satisfy certain model-theoretic tameness assumptions
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 toward this conjecture by showing that every infinite ndependent 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
Summary
A classical line of research in model theory, both pure and applied, aims to determine properties of algebraic structures, such as groups and fields, that satisfy certain model-theoretic tameness assumptions. We provide new examples of strictly 2-dependent fields with additional structure by showing that Granger’s examples of non-degenerate bilinear forms over dependent fields are strictly 2dependent in Section 6 (demonstrating in particular the necessity of the pure ring language assumption in Conjecture 1.1) Our proof of this relies on establishing some general results on -dependent theories, possibly of independent interest: a reduction of the -dependence of a theory to formulas with all but one of its variables singletons (Section 2), a type-counting criterion for 2-dependence and the Composition Lemma showing 2-dependence of compositions of dependent relations with binary functions (Section 5). Our proof for relations of higher arity relies on an infinitary generalization of Hrushovski’s observation [25] that the random -ary hypergraph is not a finite Boolean combination of relations of arity − 1
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