An official website of the United States government
null
(Ed.)
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 toward 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.
more »
« less
