The main contribution of this note is to establish a framework to extend results of tensor functions over specific field to general field. As a consequence of this framework, we extend the existing work to more general settings: \emph{(1)} slice rank is linearly bounded by geometric rank for any 3-tensors over any field. \emph{(2)} slice rank of any 3-tensors is quasi-supermultiplicative. As a consequence, the asymptotic slice rank exists for any 3-tensors.

翻译：本文的主要贡献是建立了一个框架，将特定域上的张量函数结果推广到一般域。作为该框架的推论，我们将现有工作推广至更一般的设定中：（1）在任何域上，任意3-张量的切片秩都被几何秩线性有界；（2）任意3-张量的切片秩具有拟超乘性。因此，任意3-张量的渐近切片秩存在。