Tur\'{a}n problems for multilinear maps

TL;DR

This paper extends the understanding of Turán problems for multilinear maps by determining the isotropy index over algebraically closed and large finite fields, and connects these results to classical combinatorial and tensor rank problems.

Contribution

It generalizes the isotropy index formula to multilinear maps of any order, solves an open question, and links these findings to the Feldman--Propp, Turán, and Gow--Quinlan numbers.

Findings

Extended isotropy index formula to multilinear maps of arbitrary order.

Determined exact Feldman--Propp number for algebraically closed fields.

Established exact Turán and Gow--Quinlan numbers for multilinear maps.

Abstract

This paper is concerned with Tur\'{a}n 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 Tur\'{a}n 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 Erd\H{o}s 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.