Mathematicians tighten limits on algebra growth, sharpening computational theory
Researchers have improved a mathematical bound that describes how fast certain algebraic structures can expand—a problem relevant to computational complexity and data structure design. The finding refines decades-old limits, potentially informing how engineers optimize algorithms and systems that depend on predictable growth rates.
Originaltitel: Bounds for the entropy of graded algebras
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.