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

Artem Chernikov, Nadja Hempel

TL;DR

This work develops a comprehensive model-theoretic treatment of multilinear forms, proving quantifier elimination for infinite-dimensional alternating non-degenerate $n$-linear spaces over NIP fields and establishing their $n$-dependence (strictly so in the generic case). It generalizes Granger’s bilinear results via a higher-arity Composition Lemma, combining finitary type-counting and array-shattering techniques to show that compositions preserve $n$-dependence. The paper also demonstrates NSOP$_1$ preservation in multilinear form theories and investigates invariant connected components $G^{ abla}$ in $n$-dependent abelian groups, obtaining a relative absoluteness result and applying it to finite-field multilinear form examples. These findings illuminate the tame behavior of multilinear-structured theories, connect $n$-dependence to stability-theoretic properties, and provide concrete structural descriptions of connected components in group settings.

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.

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

TL;DR

This work develops a comprehensive model-theoretic treatment of multilinear forms, proving quantifier elimination for infinite-dimensional alternating non-degenerate -linear spaces over NIP fields and establishing their -dependence (strictly so in the generic case). It generalizes Granger’s bilinear results via a higher-arity Composition Lemma, combining finitary type-counting and array-shattering techniques to show that compositions preserve -dependence. The paper also demonstrates NSOP preservation in multilinear form theories and investigates invariant connected components in -dependent abelian groups, obtaining a relative absoluteness result and applying it to finite-field multilinear form examples. These findings illuminate the tame behavior of multilinear-structured theories, connect -dependence to stability-theoretic properties, and provide concrete structural descriptions of connected components in group settings.

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 in n-dependent groups, demonstrating their relative absoluteness in the abelian case.
Paper Structure (22 sections, 35 theorems, 177 equations)

This paper contains 22 sections, 35 theorems, 177 equations.

Table of Contents

  1. Introduction
  2. Multilinear forms
  3. Perfect pairings, non-degenerate and generic $n$-linear forms
  4. Finite dimensional case
  5. Infinite dimensional case
  6. Quantifier elimination and completeness
  7. Composition Lemma for functions of arbitrary arity and NIP relations
  8. NIP formulas on indiscernible sequences and arrays
  9. $N$-dependence and generalized indiscernibles
  10. Finitary type-counting characterization of $n$-dependence
  11. Array shattering lemma in NIP structures
  12. Composition Lemma
  13. Discussion
  14. Non-degenerate $n$-linear forms are $n$-dependent and NSOP$_1$
  15. $N$-dependence
  16. ...and 7 more sections

Key Result

Theorem 1

For any $n \geq 1$ and field $K$, the theory $\prescript{}{\operatorname{Alt}}T^K_n$ of infinite dimensional alternating non-degenerate $n$-linear spaces over $K$ has quantifier elimination in the language $\mathcal{L}^{K}_{\theta,f}$, and is complete. If $K$ is finite, then $\prescript{}{\operatorn

Theorems & Definitions (111)

  • Theorem : Theorem \ref{['thm: QE for multilinear forms']}
  • Theorem : Composition Lemma, Theorem \ref{['Composition Lemma']}
  • Theorem : see Proposition \ref{['prop: type count']} for a precise statement
  • Theorem : Array Shattering Lemma, Lemma \ref{['lem: comp lemma induction']}(2) for $k=2$
  • Theorem : Theorem \ref{['thm: Granger']}
  • Theorem : Theorem \ref{['thm: NSOP1 and Kim indep']}
  • Conjecture 1.2
  • Theorem : Theorem \ref{['thm: Ginfty k-dep main']}
  • Definition 2.1
  • Definition 2.2
  • ...and 101 more