Asymptotic tensor rank is notoriously difficult to determine. Indeed, determining its value for the $2\times 2$ matrix multiplication tensor would determine the matrix multiplication exponent, a long-standing open problem. On the other hand, Strassen's asymptotic rank conjecture makes the bold claim that asymptotic tensor rank equals the largest dimension of the tensor and is thus as easy to compute as matrix rank. Despite tremendous interest, much is still unknown about the structural and computational properties of asymptotic rank; for instance whether it is computable. We prove that asymptotic tensor rank is "computable from above", that is, for any real number $r$ there is an (efficient) algorithm that determines, given a tensor $T$, if the asymptotic tensor rank of $T$ is at most $r$. The algorithm has a simple structure; it consists of evaluating a finite list of polynomials on the tensor. Indeed, we prove that the sublevel sets of asymptotic rank are Zariski-closed (just like matrix rank). While we do not exhibit these polynomials explicitly, their mere existence has strong implications on the structure of asymptotic rank. As one such implication, we find that the values that asymptotic tensor rank takes, on all tensors, is a well-ordered set. In other words, any non-increasing sequence of asymptotic ranks stabilizes ("discreteness from above"). In particular, for the matrix multiplication exponent (which is an asymptotic rank) there is no sequence of exponents of bilinear maps that approximates it arbitrarily closely from above without being eventually constant. In other words, any such upper bound on the matrix multiplication exponent that is close enough, will "snap" to it. Previously such discreteness results were only known for finite fields or for other tensor parameters (e.g., asymptotic slice rank). We obtain them for infinite fields like the complex numbers.

翻译：渐近张量秩的确定极为困难。事实上，若确定2×2矩阵乘法张量的渐近秩便能得到矩阵乘法指数，而这是一个长期未解的公开问题。另一方面，施特拉森渐近秩猜想大胆断言：渐近张量秩等于张量的最大维度，因而与矩阵秩一样易于计算。尽管学界对此高度关注，但关于渐近秩的结构与计算性质仍有许多未知——例如其是否可计算。我们证明渐近张量秩是"由上可计算的"：即对任意实数r，存在（高效）算法能判定给定张量T的渐近秩是否不超过r。该算法结构简单，仅需在张量上评估有限个多项式。事实上，我们证明渐近秩的子水平集是Zariski闭集（如同矩阵秩）。虽然未显式给出这些多项式，但其存在性已对渐近秩的结构产生深远影响。由此推出：全体张量的渐近秩取值构成良序集合——换言之，任何非增渐近秩序列必然稳定（"由上离散性"）。特别地，对于矩阵乘法指数（一种渐近秩），不存在双线性映射指数序列能任意逼近其上方且不最终恒定。即任何足够接近的矩阵乘法指数上界都将"吸附"至该指数。此前此类离散性结果仅对有限域成立，或仅适用于其他张量参数（如渐近切片秩）。现我们将其推广至复数域等无限域。