A base change framework for tensor functions

Abstract

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.