Let F be a field and let Q(alpha) = qsub1supdsub1(alpha)...qsubksupdsubk(alpha) \(mo F[alpha] be a polynomial of degree n where qsub1(alpha),...,qsubk(alpha) are distinct irreducible polynomials. Let ysub1(alpha),...,ysubr(alpha), xsub1(alpha),...,xsub3(alpha) be $r~+~s,~n~-~$1-degree polynomials.
It is shown that if Card(F) >= maxsub1<=i<=k 2deg qsubisupdsubi(alpha) − 2 then the number of nonscalar multiplications/divisions required to compute the coefficients of xsubi(alpha)ysub1(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.