Tur\'{a}n problems for multilinear maps

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.