On n-dependent groups and fields III. Multilinear forms and invariant connected components
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913765109268480 |
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| author | Chernikov, Artem Hempel, Nadja |
| author_facet | Chernikov, Artem Hempel, Nadja |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_19921 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On n-dependent groups and fields III. Multilinear forms and invariant connected components Chernikov, Artem Hempel, Nadja Logic Combinatorics Group Theory 03C45, 03C60, 05C65, 05C35, 15A69, 47A07 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. |
| title | On n-dependent groups and fields III. Multilinear forms and invariant connected components |
| topic | Logic Combinatorics Group Theory 03C45, 03C60, 05C65, 05C35, 15A69, 47A07 |
| url | https://arxiv.org/abs/2412.19921 |