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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
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, vol. 00, no. , pp. 224-234, 2012, doi:10.1109/CCC.2012.13
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