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

2024/12/27 by Artem Chernikov, Chernikov, Artem, Nadja Hempel +1 · 1 citation
Mathematics · #03C45 #03C60 #05C35 #05C65 #15A69 #47A07 #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2412.19921

openalex publication_date 2024/12/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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^∞ in n-dependent groups, demonstrating their relative absoluteness in the abelian case.

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