Thomas Karam

The structure of the set of minimal-length slice rank decompositions of a tensor over a finite field.

If k is a nonnegative integer and F is a field, then all length-k decompositions of a matrix with rank k over F can be obtained from one another after a change of basis. In particular, if the field F is finite, then the number of such decompositions can be expressed in terms of k and F, and is bounded above by |F|^{k^2}. More generally, if d => 2 is an integer, then the number of length-k tensor rank decompositions of an order-d tensor with tensor rank k is bounded above by |F|^{(d-1)k^2}. We will explain how an analogous statement can be obtained for the slice rank: although it is no longer true that the number of length-k slice rank decompositions of an order-d tensor with slice rank k is bounded above in terms of d,k and |F|, this is nonetheless the case up to a class of transformations which can be described in a simple way.