Beyond the Alder-Strassen bound

We also deal with the complexity of multiplication in algebras with nonzero radical. We present an example that shows that our methods in the semisimple case cannot be applied directly to this problem. We exhibit lower bound techniques for C(A) that yield bounds still significantly above the Alder-Strassen bound. The main application is the lower bound C(Tn(k))⩾(218-o(1))dimTn(k) for the multiplicative complexity of multiplication in the algebra Tn(k) of upper triangular n×n-matrices.