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ON THE COMPLEXITY OF BILINEAR FORMS OVER ASSOCIATIVE ALGEBRAS

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Author
Bshouty, Nader H.
Accessioned
2008-05-26T20:40:33Z
Available
2008-05-26T20:40:33Z
Computerscience
1999-05-27
Issued
1989-12-01
Subject
Computer Science
Type
unknown
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Abstract
Let $F$ be a field and let $Q( alpha )~=~q sub 1 sup {d sub 1} ( alpha )... q sub k sup {d sub k} ( alpha )~\(mo~F[ alpha ]$ be a polynomial of degree $n$ where $q sub 1 ( alpha ),...,q sub k ( alpha )$ are distinct irreducible polynomials. Let $y sub 1 ( alpha ),...,y sub r ( alpha ),~x sub 1 ( alpha ),...,x sub 3 ( alpha )$ be $r~+~s,~n~-~$1-degree polynomials. It is shown that if $Card(F)~>=~max sub {1 <=i<=k}~2deg~q sub i sup d sub i ( alpha )~-~2$ then the number of nonscalar multiplications/divisions required to compute the coefficients of $x sub i ( alpha ) y sub 1 ( alpha )~ mod~Q( alpha ),~i=1,...,s$ by straight line algorithms is $ s(2n~-~k) $. We also prove that if $H$ is an $s~times~r$ matrix with entries from $F$ then the number of nonscalar multiplications/divisions required to compute the coefficients of $(x sub 1 ( alpha ),...,x sub s ( alpha ))H(y sub 1 ( alpha ), ...,y sub r ( alpha )) sup T $ by straight line algorithms is $ rank(H)(2n~-~k)$. All those systems satisfy the direct sum conjecture strongly. For some other algebras that are direct sum of local algebras the above results also hold.
Notes
We are currently acquiring citations for the work deposited into this collection. We recognize the distribution rights of this item may have been assigned to another entity, other than the author(s) of the work.If you can provide the citation for this work or you think you own the distribution rights to this work please contact the Institutional Repository Administrator at digitize@ucalgary.ca
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University of Calgary
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Science
Doi
http://dx.doi.org/10.11575/PRISM/30484
Uri
http://hdl.handle.net/1880/46600
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