On the Complexity of Bilinear Forms over Associative Algebras

Abstract

Let F be a field, and let $q(\alpha ) = q_1^{d_1 } (\alpha ) \cdots q_1^{d_k } (\alpha ) \in F[\alpha ]$ be a polynomial of degree n, where $q_1 (\alpha ) \cdots q_k (\alpha )$ are distinct irreducible polynomials. Let $y(\alpha ),y_1 (\alpha ), \ldots ,y_r (\alpha ),x_1 (\alpha ), \ldots ,x_s (\alpha )$ be $(n - 1)$-degree polynomials with distinct nonscalar coefficients. The authors show the following: the number of nonscalar multiplications/divisions required to compute the coefficients of $x_1 (\alpha ),y(\alpha )\bmod q(\alpha )$ for $i = 1, \ldots ,s$ by straight line algorithms is $s(2n - k)$. If H is a $s \times r$- matrix with entries from F, then the number of nonscalar multiplications/ divisions required to compute the coefficients of $(x_1 (\alpha ), \ldots ,x_s (\alpha ))H(y_1 (\alpha ), \ldots ,y_r (\alpha ))^T \bmod q(\alpha )$ by straight line algorithms is equal to $(2n - k)$ rank $(H)$. All the above systems satisfy the direct sum conjecture strongly. The above results also hold for some other algebras that are direct sums of local algebras, such as commutative algebras and division algebras.