The Border Rank of the Multiplication of 2× 2 Matrices Is Seven

One of the leading problems of algebraic complexity theory is matrix multiplication. The näıve multiplication of two n× n matrices uses n multiplications. In 1969 Strassen [19] presented an explicit algorithm for multiplying 2 × 2 matrices using seven multiplications. In the opposite direction, Hopcroft and Kerr [11] and Winograd [20] proved independently that there is no algorithm for multiplying 2× 2 matrices using only 6 multiplications. The precise number of multiplications needed to execute matrix multiplication (or any given bilinear map) is called the rank of the bilinear map. A related problem is to determine the border rank of matrix multiplication (or any given bilinear map). Roughly speaking, some bilinear maps may be approximated with arbitrary precision by less complicated bilinear maps and the border rank of a bilinear map is the complexity of arbitrarily small “good” perturbations of the map. These perturbed maps can give rise to fast exact algorithms for matrix multiplication, see [5]. The border rank made appearances in the literature in the 1980’s and early 90’s, see, e.g., [18, 7, 13, 2, 9, 10, 4, 3, 15, 14, 17, 1, 8], but to our knowledge there has not been much progress on the question since then. More precisely, for any complex projective variety X ⊂ CP = PV , and point p ∈ PV , define the X-rank of p to be the smallest number r such that p is in the linear span of r points of X. Define σr(X), the r-th secant variety of X, to be the Zariski closure of the set of points of X-rank r, and define the X-border rank of p to be the smallest r such that p ∈ σr(X). The terminology is motivated by the case X = Seg(P × P) ⊂ P(C ⊗ C), the Segre variety of rank one matrices. Then the X-rank of a matrix is just its usual rank. Let A, B, C be vector spaces and let f : A × B → C be a bilinear map, i.e., an element of A⊗B⊗C. Let X = Seg(PA × PB × PC) ⊂ P(A⊗B⊗C) denote the Segre variety of decomposable tensors in A⊗B⊗C. The border rank of a bilinear map is its X-border rank. While for the Segre product of two projective spaces, border rank coincides with rank, here they can be quite different. In this paper we prove the theorem stated in the title. Let MMult ∈ C ⊗C ⊗C denote the matrix multiplication operator for two by two matrices. Strassen [17] showed that MMult / ∈ σ5(P 3 × P × P). Our method of proof is to decompose σ6(P 3 × P × P)\σ5(P 3 × P × P) into fifteen components, and then using case by case arguments show MMult is not in any of the components. The decomposition rests upon a differential-geometric understanding of curves in submanifolds, which is carried out in §2. In §3 we roughly describe the components of σr(X) for an arbitrary variety, and give a precise description for σ6(P 3 × P × P), giving a normal form for a point of each component. Finally in §4 we carry out our case by case analysis.