A base change framework for tensor functions

TL;DR

This paper introduces a framework for extending tensor function results across different fields, leading to new bounds and properties for 3-tensors such as linear bounds on slice rank and the existence of asymptotic slice rank.

Contribution

It establishes a general framework to extend tensor function results over various fields, enabling broader applicability and new bounds for 3-tensors.

Findings

Slice rank is linearly bounded by geometric rank for any 3-tensor over any field.

Slice rank of any 3-tensor is quasi-supermultiplicative.

Asymptotic slice rank exists for any 3-tensor.

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.