Computing lower bounds on tensor rank over finite fields

摘要

A lower bound on rank is constructed for arbitrary tensors over finite fields. For fields of low cardinality the bound is more precise .than those generated by previously known techniques because the structure of the field is exploited. In addition, the proof technique over Z2 leads to a method for determining whether the lower bound constructed also represents an upper bound and, hence, the rank. As an application of this idea, it is shown that eight multiplications are necessary and sufficient to calculate the four-dimensional quaternion product over Z2.