Isotropy and completeness indices of multilinear maps

TL;DR

This paper introduces the isotropy and completeness indices for multilinear maps, providing bounds, applications to tensor ranks, a Ramsey-type theorem, and a polynomial-time algorithm for polynomial ideal height estimation.

Contribution

It defines two new invariants for multilinear maps, establishes bounds, and applies them to tensor rank bounds, a Ramsey theorem, and polynomial ideal analysis.

Findings

Bounds on indices in terms of tensor ranks and height

Resolved an open problem on tensor subrank bounds

Developed a polynomial-time algorithm for polynomial ideal height estimation

Abstract

Structures of multilinear maps are characterized by invariants. In this paper we introduce two invariants, named the isotropy index and the completeness index. These invariants capture the tensorial structure of the kernel of a multilinear map. We establish bounds on both indices in terms of the partition rank, geometric rank, analytic rank and height, and present three applications: 1) Using the completeness index as an interpolator, we establish upper bounds on the aforementioned tensor ranks in terms of the subrank. This settles an open problem raised by Kopparty, Moshkovitz and Zuiddam, and consequently answers a question of Derksen, Makam and Zuiddam. 2) We prove a Ramsey-type theorem for the two indices, generalizing a recent result of Qiao and confirming a conjecture of his. 3) By computing the completeness index, we obtain a polynomial-time probabilistic algorithm to estimate the height of a polynomial ideal.