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.

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关于 n 依赖群和域 II

我们继续研究n- 依赖的群体、领域和相关结构，很大程度上是由以下猜想推动的，即每个n-dependent 字段是相关的。我们通过证明每个无限n- 正特征的依赖值域是亨斯式的，在这种情况下获得了 Shelah 的亨斯式猜想的一个变体，并推广了 Johnson 最近对依赖场的结果。此外，我们证明了类型可定义的连接组件在通用参数集上的交集的结果n-依赖群，概括了 Shelah 的绝对性 $G^{00}$ 在依赖理论和相对绝对性 $G^{00}$ 在 $2$ 依赖理论。为了澄清这个猜想的范围，我们提供了严格的新例子 $2$ -具有附加结构的依赖域，表明格兰杰在依赖域上的非退化双线性形式的示例是 $2$ -依赖。在此过程中，我们获得了一些具有独立兴趣的纯模型理论结果：我们表明n-除了一个变量单例外，所有公式都见证了依赖性；提供一个类型计数标准 $2$ -依赖并用它来推断 $2$ - 依赖关系与任意二元函数的组合的依赖关系（组合引理）；并证明几何理论的扩展吨由泛型谓词依赖当且仅当它是n-依赖于某些人n, 当且仅当代数闭包吨被解体。Martin Bays 的附录提供了 Kaplan-Scanlon-Wagner 定理中的显式同构。