# Tensor rank is not multiplicative under the tensor product

## TL;DR

This paper demonstrates that tensor rank does not always multiply under tensor products, providing a counterexample to a previously assumed property and exploring related concepts like border rank and algebraic complexity.

## Contribution

It proves that tensor rank is not generally multiplicative under tensor products, answering a question posed by Draisma and Saptharishi.

## Key findings

- Tensor rank is not multiplicative under tensor product.
- Border rank can be strictly smaller than tensor rank.
- Lower bounds on border rank multiply under tensor product.

## Abstract

The tensor rank of a tensor t is the smallest number r such that t can be decomposed as a sum of r simple tensors. Let s be a k-tensor and let t be an l-tensor. The tensor product of s and t is a (k + l)-tensor. Tensor rank is sub-multiplicative under the tensor product. We revisit the connection between restrictions and degenerations. A result of our study is that tensor rank is not in general multiplicative under the tensor product. This answers a question of Draisma and Saptharishi. Specifically, if a tensor t has border rank strictly smaller than its rank, then the tensor rank of t is not multiplicative under taking a sufficiently hight tensor product power. The "tensor Kronecker product" from algebraic complexity theory is related to our tensor product but different, namely it multiplies two k-tensors to get a k-tensor. Nonmultiplicativity of the tensor Kronecker product has been known since the work of Strassen. It remains an open question whether border rank and asymptotic rank are multiplicative under the tensor product. Interestingly, lower bounds on border rank obtained from generalised flattenings (including Young flattenings) multiply under the tensor product.