2012 IEEE 27th Conference on Computational Complexity (2012)

Porto Portugal

June 26, 2012 to June 29, 2012

ISSN: 1093-0159

ISBN: 978-0-7695-4708-4

pp: 224-234

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/CCC.2012.13

ABSTRACT

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.

INDEX TERMS

associative algebra, algebraic complexity theory, complexity of bilinear problems

CITATION

"Algebras of Minimal Multiplicative Complexity,"

*2012 IEEE 27th Conference on Computational Complexity(CCC)*, Porto Portugal, 2012, pp. 224-234.

doi:10.1109/CCC.2012.13