Computational hardness of rank-1 wildcard decomposition when the field is not fixed

Determine the computational complexity of the rank-1 tensor decomposition problem with wildcard entries when the finite field is not fixed and varies as F = F_{p^k}, specifically establishing whether the problem remains polynomial-time solvable or becomes computationally hard as p and k vary.

Background

For a fixed finite field, the paper outlines a polynomial-time approach to rank-1 wildcard decomposition by reducing multiplicative constraints to linear relations via discrete logarithms. However, an efficient algorithm to find a primitive element in a general finite field is not known, and the input size depends on p and k in F_{p^k}, making the complexity unclear when the field is part of the input.