Improvements of the Alder-Strassen bound: Algebras with nonzero radical

Abstract

Let C(A) denote the multiplicative complexity of a finite dimensional associative k-algebra A. For algebras A with nonzero radical rad A, we exhibit several lower bound techniques for C(A) that yield bounds significantly above the AlderStrassen bound. In particular, we prove that the multiplicative complexity of the multiplication in the algebras κ is bounded from below by 3 where I denotes the ideal generated by all monomials of degree d in X 1 Furthermore, we show the lower bound C(Tn(k)) ≥ (2 1/8 o(1)) dimTn(k) for the multiplication of upper triangular matrices. © 2011 Springer-Verlag Berlin Heidelberg.