Bounds for Geometric rank in Terms of Subrank (2506.16132v1)

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.