On n-dependent groups and fields III. Multilinear forms and invariant connected components

TL;DR

This paper advances the model theory of multilinear forms, establishing quantifier elimination and analyzing dependence properties of associated fields and groups, including invariant connected components and their absoluteness.

Contribution

It generalizes previous bi-linear results to multilinear forms, introduces a Composition Lemma for arbitrary arity functions, and explores the dependence properties of fields and groups in this context.

Findings

The theory of infinite dimensional non-degenerate alternating n-linear spaces over an NIP field is strictly n-dependent.

The theory is NSOP1 if the base field is NSOP1.

Invariant connected components in n-dependent groups are relatively absolute in the abelian case.

Abstract

We develop some model theory of multi-linear forms, generalizing Granger in the bi-linear case. In particular, after proving a quantifier elimination result, we show that for an NIP field K, the theory of infinite dimensional non-degenerate alternating n-linear spaces over K is strictly n-dependent; and it is NSOP1 if K is. This relies on a new Composition Lemma for functions of arbitrary arity and NIP relations (which in turn relies on certain higher arity generalizations of Sauer-Shelah lemma). We also study the invariant connected components $G^{\infty}$ in n-dependent groups, demonstrating their relative absoluteness in the abelian case.