Ranks and another problem · [Mar 9, 08:49 AM]
It turned out that my conjecture only holds for commutative variables. The problem can be reduced to finding the so called border rank on the space of symmetric tensors. For arbitrary order 2 tensors (a homogeneous noncommuting polynomial ) the number of products required is simply the rank of it as a matrix. The so called lower rank approximations are apparently quite important in image processing and the like.
I think matrix rank of desnity matrices can be used to define some properties also as I vaguely remember a paper; not a profound formalism but useful I guess.
Now I have the following question [which I haven’t spend much time on]:
Solve the following recursion, asymptotically for large n:d(n+1)=binomial[n+c(n),c(n)-1]
c(n+1)=d(n+1) lg(n)
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