Isotropy and completeness indices of multilinear maps
Qiyuan Chen, Ke Ye
TL;DR
The article defines two central invariants for $\mathsf{K}$-multilinear maps, the isotropy index $\alpha(F)$ and the completeness index $\beta(F)$, and anchors them within the known tensor-rank framework via the associated tensor $T_F$. It proves two core bounds that relate $\alpha(F)$ to partition rank and, dually, $\beta(F)$ to geometric/analytic ranks and height, enabling a unified view of decomposition versus randomness in multilinear maps. Three key applications follow: (i) interpolating tensor ranks through subrank to resolve open problems; (ii) a Ramsey-type theory for isotropy and completeness indices across multilinear maps; and (iii) a polynomial-time probabilistic algorithm to estimate the height of polynomial ideals from these indices. Together, the results connect combinatorial, algebraic, and computational facets of multilinear maps, offering new bounds, algorithmic tools, and extensions to higher-order tensors.
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.