Asymptotic rank bounds: a numerical census
Kisun Lee
TL;DR
The paper addresses bounding the asymptotic rank of tensor powers using secant-variety geometry via the K_q framework. It combines numerical implicitization with monodromy-based sampling of L ∩ σ_r(V) to certify nonvanishing of degree-q polynomials and thus upper bounds on lim_{q→∞} rank(T^⊗q)^{1/q}. Across formats with generic border rank ≤ 20, it reports 5 cases with strictly improved bounds and provides numerical data toward Strassen's asymptotic rank conjecture while clarifying numeric barriers in higher codimension. The work emphasizes the connection between algebraic geometry and numerical methods and raises open questions about when improvements are possible and what advances are needed to extend the results.
Abstract
We systematically compute improved asymptotic rank bounds for tensors. Using numerical implicitization, we implement the geometric framework of Kaski and Michałek across all computationally feasible cases. By detecting the absence of low-degree vanishing polynomials on secant varieties, we obtain new asymptotic rank bounds that improve upon the generic border rank bounds. The results provide numerical data supporting Strassen's asymptotic rank conjecture and clarify the computational barriers posed by current numerical methods.