Bounds for Geometric rank in Terms of Subrank

Abstract

For tensors of fixed order, we establish three types of upper bounds for the geometric rank in terms of the subrank. Firstly, we prove that, under a mild condition on the characteristic of the base field, the geometric rank of a tensor is bounded by a function in its subrank in some field extension of bounded degree. Secondly, we show that, over any algebraically closed field, the geometric rank of a tensor is bounded by a function in its subrank. Lastly, we prove that, for any order three tensor over an arbitrary field, its geometric rank is bounded by a quadratic polynomial in its subrank. Our results have several immediate but interesting implications: (1) We answer an open question posed by Kopparty, Moshkovitz and Zuiddam concerning the relation between the subrank and the geometric rank; (2) For order three tensors, we generalize the Biaggi-Chang- Draisma-Rupniewski (resp. Derksen-Makam-Zuiddam) theorem on the growth rate of the border subrank (resp. subrank), in an optimal way; (3) For order three tensors, we generalize the Biaggi- Draisma-Eggleston theorem on the stability of the subrank, from the real field to an arbitrary field; (4) We confirm the open problem raised by Derksen, Makam and Zuiddam on the maximality of the gap between the subrank of the direct sum and the sum of subranks; (5) We derive, for the first time, a de-bordering result for the border subrank and upper bounds for the partition rank and analytic rank in terms of the subrank; (6) We reprove a gap result for the subrank.