11,674 research outputs found
New lower bounds for the rank of matrix multiplication
The rank of the matrix multiplication operator for nxn matrices is one of the
most studied quantities in algebraic complexity theory. I prove that the rank
is at least n^2-o(n^2). More precisely, for any integer p\leq n -1, the rank is
at least (3- 1/(p+1))n^2-(1+2p\binom{2p}{p-1})n. The previous lower bound, due
to Blaser, was 5n^2/2-3n (the case p=1).
The new bounds improve Blaser's bound for all n>84. I also prove lower bounds
for rectangular matrices significantly better than the the previous bound.Comment: Completely rewritten, mistake in error term in previous version
corrected. To appear in SICOM
A note on the gap between rank and border rank
We study the tensor rank of the tensor corresponding to the algebra of
n-variate complex polynomials modulo the dth power of each variable. As a
result we find a sequence of tensors with a large gap between rank and border
rank, and thus a counterexample to a conjecture of Rhodes. At the same time we
obtain a new lower bound on the tensor rank of tensor powers of the generalised
W-state tensor. In addition, we exactly determine the tensor rank of the tensor
cube of the three-party W-state tensor, thus answering a question of Chen et
al.Comment: To appear in Linear Algebra and its Application
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