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Bounds for the Entropy of Graded Algebras

Jan Snellman
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Abstract

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.

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Jan Snellman. 2002-09-07. Bounds for the Entropy of Graded Algebras. https://doi.org/10.1080/00927872.2026.2665319

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