TITLE: Strassen Algorithm 7 multiplications
QUESTION [9 upvotes]: Strassen Algoritm is a well-known matrix multiplication divide and conquer algorithm.
The trick of the algorithm is reducing the number of multiplications to 7 instead of 8. I was wondering, can we reduce any further? Can we only do 6 multiplications?
Also, what happens if we divide the NxN arrays into 9 arrays each of (N/3)x(N/3) instead of 4 arrays of (N/2)x(N/2). Can we then do less multiplications?

REPLY [2 votes]: Of possible relevance to your question (though it might go into more geometric considerations than you care to read):
Generalizations of Strassen's equations for secant varieties of Segre varieties (J. Landsberg and  L. Manivel, Comm. Algebra 2008)