Exact rank of the matrix multiplication tensor

Determine the exact tensor rank of the matrix multiplication tensor for matrices of sizes n×k and k×m over a field K, i.e., determine the minimal number of rank-one tensors whose sum equals the tensor in K^{n×k}⊗K^{k×m}⊗K^{n×m} defined by Σ_{u=1}^n Σ_{v=1}^k Σ_{w=1}^m a_{u,v}⊗b_{v,w}⊗c_{u,w}.

Background

The paper frames matrix multiplication as a tensor in K^{n×k}⊗K^{k×m}⊗K^{n×m} and highlights that the efficiency of matrix multiplication algorithms corresponds to finding low-rank decompositions of this tensor. While asymptotic bounds via the exponent ω are known, the exact rank for concrete sizes remains unknown.

This open problem is foundational in algebraic complexity theory: an exact rank characterization would directly translate to optimal algorithms with explicit multiplication counts for given small dimensions and inform the broader pursuit of minimizing ω. The authors mention this to contrast with the more tractable case of polynomial multiplication explored in the paper.