The slice-rank method, introduced by Tao as a symmetrized version of the polynomial method of Croot, Lev and Pach and Ellenberg and Gijswijt, has proved to be a useful tool in a variety of combinatorial problems. Explicit tensors have been introduced in different contexts but little is known about the limitations of the method.

In this paper, building upon a method presented by Tao and Sawin, it is proved that the asymptotic slice rank of any k-tensor in any field is either 1 or at least $k/{(k-1)}^{(k-1)/k}$. This provides evidence that straight-forward application of the method cannot give useful results in certain problems for which non-trivial exponential bounds are already known. An example, actually a motivation for starting this work, is the problem of bounding the size of sets of trifferent sequences, which constitutes a long-standing open problem in information theory and in theoretical computer science.

Tao提出的切片秩方法是Croot，Lev和Pach以及Ellenberg和Gijswijt多项式方法的对称版本，已被证明是解决各种组合问题的有用工具。在不同的上下文中引入了显式张量，但对该方法的局限性知之甚少。

本文基于Tao和Sawin提出的方法，证明了在任何场中任何k张量的渐近切片秩为1或至少$ķ/{（ķ-\mathrm{1个}）}^{（ķ-\mathrm{1个}）/ķ}$。这提供了证据，表明该方法的直接应用无法在某些已知非平凡指数范围的问题中给出有用的结果。一个例子，实际上是开始这项工作的动力，是限制不同序列集的大小的问题，它构成了信息论和理论计算机科学中一个长期存在的开放性问题。