Polynomial-time boundary for rank-R tensor decomposition (R ≥ 5)

Determine the set of integers R ≥ 5 for which the decision problem of finding a rank-R decomposition of an n×n×n tensor over a fixed finite field (with no wildcard entries) can be solved in polynomial time with respect to n.

Background

The paper proves a polynomial-time algorithm for finding rank-R decompositions of 3D tensors over a fixed finite field when R ≤ 4. Extending such polynomial-time guarantees to higher ranks remains unresolved and is central to understanding the tractability frontier of low-rank tensor decomposition over finite fields.