Newman, Schneider and Shalev defined the entropy of a graded associative algebra A as H(A) = \limsup_{ n \to \infty} \sqrt[n]{ a_n}, where a_n is the vector space dimension of the n'th homogeneous component. When A is the homogeneous quotient of a finitely generated free associative algebra, they showed that H(A) \le \sqrt{ a_2}. Using some results of Friedland on the maximal spectral radius of 0-1 matrices with a prescribed number of ones, we improve on this bound.
Categories: math.RA