Turán problems for multilinear maps

This paper is concerned with Turán problems for (alternating) multilinear maps, with the aim of determining the maximum dimension $k$, called the isotropy index, for which every such map has an isotropic subspace of dimension $k$. We extend the formula for the isotropy index of alternating bilinear maps [Buhler, Gupta & Harris, J. Algebra, 1987] to alternating multilinear maps of arbitrary order over algebraically closed fields. In particular, this answers an open question posed in [Qiao, Discrete Anal., 2023]. Moreover, we prove that the same formula holds for sufficiently large finite fields. For multilinear maps, we establish a necessary and sufficient condition for the isotropy index to be at least two. Our results have three implications: (1) For algebraically closed fields, we determine the exact value of the Feldman–Propp number, whose lower bound has been known for over thirty years [Feldman & Propp, Adv. Math., 1992] but whose precise value had remained undetermined. (2) We establish the exact values of both the Turán number and the Gow–Quinlan number for alternating multilinear maps of arbitrary order over algebraically closed fields. Specifically, our result greatly extends the existing formula for the Gow–Quinlan number for alternating bilinear maps [Gow & Quinlan, Linear Multilinear Algebra, 2006]. (3) Bridging the lower bound for the Erdős box problem and tensor analytic rank, we show that there is an obstruction to improving the existing lower bound in [Conlon, Pohoata & Zakharov, Discrete Anal., 2021] via the multilinear method.

💡 Research Summary

The paper studies Turán-type extremal problems for alternating multilinear maps, focusing on the isotropy index – the largest dimension k such that every map admits a k‑dimensional isotropic subspace. For alternating bilinear maps (order 2) the isotropy index was known from the work of Buhler, Gupta and Harris (1987). The authors extend this to alternating maps of arbitrary order d over algebraically closed fields and, remarkably, show that the same formula holds for sufficiently large finite fields.

The main result (Theorem 1.2) states that for an algebraically closed field F, integers n, d, m with m≥2, \