2012 IEEE 27th Conference on Computational Complexity (2012)
June 26, 2012 to June 29, 2012
DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/CCC.2012.13
We prove that an associative algebra $A$ has minimal rank if and only if the Alder -- Strassen bound is also tight for the multiplicative complexity of $A$, that is, the multiplicative complexity of $A$ is $2 \dim A - t_A$ where $t_A$ denotes the number of maximal two sided ideals of $A$. This generalizes a result by E. Feig who proved this for division algebras. Furthermore, we show that if $A$ is local or super basic, then every optimal quadratic computation for $A$ is almost bilinear.
associative algebra, algebraic complexity theory, complexity of bilinear problems
"Algebras of Minimal Multiplicative Complexity," 2012 IEEE 27th Conference on Computational Complexity(CCC), Porto Portugal, 2012, pp. 224-234.