Improvements of the Alder-Strassen Bound: Algebras with Nonzero Radical

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 Alder-Strassen bound. In particular, we prove that the multiplicative complexity of the multiplication in the algebras k[X1, ..., Xn]/ Id+1(X1, ..., Xn) is bounded from below by 3ċ(n+d/n) - (n+ċd/2ċ/n) - (n+ċd/2ċ/n), where Id(X1, ..., Xn) denotes the ideal generated by all monomials of degree d in X1, ..., Xn. Furthermore, we show the lower bound C(Tn(k)) ≥ (2 1/8 - o(1)) dim Tn(k) for the multiplication of upper triangular matrices.