Multiplicative Complexity and Algebraic Structure

摘要

The classical structure theory of an (associative unitary) algebra A over a field F is invoked to determine upper bounds on the (bilinear) multiplicative complexity π(A) of A over F. The upper bound problem for matrix multiplication over a finite extension F of the rational numbers is related to the multiplicative complexity problem for a certain twisted polynomial algebra. For certain base fields F (including finite extensions of the rationals and the real and complex fields) the order of complexity of an F-algebra with all nilpotent ideals having square zero is shown to be bounded above by the complexity of multiplying matrices over F.