This question is related to a previous unanswered question of mine.  Please see that question for background.When Hstad proved that computing tensor rank is NP-complete, the rank in question was allowed to be part of the input.  Specifically, given a Boolean formula with $n$ variables and $m$ clauses, Hstad constructed a tensor that has rank exactly $4n+2m$ if the formula is satisfiable and has some larger rank otherwise.What if we fix the bound on the rank to some constant?  For example...What is the complexity of deciding if a tensor has rank at most 3?The tensor should be defined over an infinite field, otherwise the problem is trivial (...only a finite number of things to check).  Answers for the natural infinite fields $\mathbb{Q}$, $\mathbb{R}$, and $\mathbb{C}$ would be best.  I am personally interested in this question using the complex numbers.
Deciding bound on Tensor Rank for a fixed value
ds.algorithms;reference request;tensor rank