New lower bounds for matrix multiplication and the 3x3 determinant
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Let M_〈 u,v,w〉∈ C^uv⊗ C^vw⊗ C^wu denote the matrix multiplication tensor (and write M_n=M_〈 n,n,n〉) and let det_3∈ ( C^9)^⊗ 3 denote the determinant polynomial considered as a tensor. For a tensor T, let R(T) denote its border rank. We (i) give the first hand-checkable algebraic proof that R(M_2)=7,(ii) prove R(M_〈 223〉)=10, and R(M_〈 233〉)=14, where previously the only nontrivial matrix multiplication tensor whose border rank had been determined was M_2,(iii) prove R( M_3)≥ 17, (iv) prove R( det_3)=17, improving the previous lower bound of 12, (v) prove R(M_〈 2nn〉)≥ n^2+1.32n for all n≥ 25 (previously only R(M_〈 2nn〉)≥ n^2+1 was known) as well as lower bounds for 4≤ n≤ 25, and (vi) prove R(M_〈 3nn〉)≥ n^2+2 n+1 for all n≥ 21, where previously only R(M_〈 3nn〉)≥ n^2+2 was known, as well as lower boundsfor 4≤ n≤ 21. Our results utilize a new technique initiated by Buczyńska and Buczyński, called border apolarity. The two key ingredients are: (i) the use of a multi-graded ideal associated to a border rank r decomposition of any tensor, and (ii) the exploitation of the large symmetry group of T to restrict to B_T-invariant ideals, where B_T is a maximal solvable subgroup of the symmetry group of T.
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