Improvements of the Alder-Strassen Bound: Algebras with Nonzero Radical
نویسنده
چکیده
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[X1; : : : ; Xn]= Id+1 (X1; : : : ; Xn) is bounded from below by 3 n+d n n+dd=2e n n+bd=2c 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 18 o(1)) dimTn(k) for the multiplication of upper triangular matrices.
منابع مشابه
Semisimple Algebras of Almost Minimal Rank over the Reals
A famous lower bound for the bilinear complexity of the multiplication in associative algebras is the Alder–Strassen bound. Algebras for which this bound is tight are called algebras of minimal rank. After 25 years of research, these algebras are now well understood. We here start the investigation of the algebras for which the Alder–Strassen bound is off by one. As a first result, we completel...
متن کاملBounds for Bilinear Complexity of Noncommutative Group Algebras
We study the complexity of multiplication in noncommutative group algebras which is closely related to the complexity of matrix multiplication. We characterize such semisimple group algebras of the minimal bilinear complexity and show nontrivial lower bounds for the rest of the group algebras. These lower bounds are built on the top of Bläser’s results for semisimple algebras and algebras with ...
متن کاملAlgebras of Minimal Rank over Arbitrary Fields
Let R(A) denote the rank (also called bilinear complexity) of a nite dimensional associative algebra A. A fundamental lower bound for R(A) is the so-called Alder{Strassen bound R(A) 2 dim A?t, where t is the number of maximal twosided ideals of A. The class of algebras for which the Alder{Strassen bound is sharp, the so-called algebras of minimal rank, has received a wide attention in algebraic...
متن کاملAlgebras of Minimal Rank over Perfect Fields
Let R(A) denote the rank (also called bilinear complexity) of a finite dimensional associative algebra A. A fundamental lower bound for R(A) is the so-called Alder– Strassen boundR(A) 2 dimA t, where t is the number of maximal twosided ideals of A. The class of algebras for which the Alder–Strassen bound is sharp, the so-called algebras of minimal rank, has received a wide attention in algebrai...
متن کامل