Asymptotic rank bounds: a numerical census

Abstract

We systematically compute improved asymptotic rank bounds for tensors. Using numerical implicitization, we implement the geometric framework of Kaski and Micha{\l}ek 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.