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arXiv · 2412.19921

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

Artem Chernikov·Nadja Hempel
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Abstract

We develop some model theory of multilinear forms, generalizing Granger's work in the bilinear 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 is NSOP$_1$ if $K$ is. These results rely on a new Composition Lemma for functions of arbitrary arity and NIP relations (which in turn relies on certain higher-arity generalizations of the Sauer--Shelah lemma). We also study the invariant connected components $G^{\infty}$ in $n$-dependent groups, demonstrating their relative absoluteness.

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Artem Chernikov, Nadja Hempel. 2026-09-18. On n-dependent groups and fields III. Multilinear forms and invariant connected components. https://arxiv.org/abs/2412.19921

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