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.

翻译：多重线性映射的结构由不变量刻画。本文引入两个不变量，分别命名为各向同性指数与完备性指数。这些不变量捕捉了多重线性映射核的张量结构。我们建立了这两个指数关于划分秩、几何秩、解析秩和高度的界，并给出三个应用：1）以完备性指数作为插值工具，我们建立了前述张量秩关于子秩的上界。这解决了Kopparty、Moshkovitz和Zuiddam提出的公开问题，并由此回答了Derksen、Makam和Zuiddam的问题。2）我们证明了这两个指数的Ramsey型定理，推广了Qiao的最新结果并证实了他的猜想。3）通过计算完备性指数，我们获得了一个多项式时间概率算法来估计多项式理想的高度。