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 radA, 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[X ₁,..., X _n/I ^d+¹ (X ₁,..., X _n) is bounded from below by $3. \left( \begin{gathered} n + d \hfill \\ n \hfill \\ \end{gathered} \right) - \left( \begin{gathered} n + \left[ {d/2} \right] \hfill \\ n \hfill \\ \end{gathered} \right) - \left( \begin{gathered} n + \left[ {d/2} \right] \hfill \\ n \hfill \\ \end{gathered} \right),$ where I _d (X ₁,..., X _n) denotes the ideal generated by all monomials of degree d in X ₁,...,X _n. Furthermore, we show the lower bound C(T _n (k) ≥ (2 1/8 - o(1)) dim Tⁿ(k) for the multiplication of upper triangular matrices.