Bivariate polynomial multiplication

We study the multiplicative complexity and the rank of the multiplication in the local algebras R/sub m,n/=k[x,y]/(x/sup m+1/,y/sup n+1/) and T/sub n/=k[x,y]/(x/sup n+1/,x/sup n/y,...,y/sup n+1/) of bivariate polynomials. We obtain the lower bounds (21/3-0(1))/spl middot/dim R/sub m,n/, and (2 1/2 -0(1))/spl middot/dim T/sub n/ for the multiplicative complexity of the multiplication in R/sub m,n/ and T/sub n/, respectively. On the other hand, we derive the upper bounds 3/spl middot/dim T/sub n/-2n-2 and 3/spl middot/dim R/sub m.n/-m-n-3 for the rank of the multiplication in T/sub n/ and R/sub m,n/, respectively, provided that the ground field k admits "fast" univariate polynomial multiplication mod x/sup N/-1. Our results are also applicable to arbitrary finite dimensional algebras of truncated bivariate polynomials k[x,y]/I, where the ideal I=(x(d/sub 0/+1),x(d/sub 1/+1)y,...,x(d/sub n/+1)y/sup n/,y/sup n+1/) is described by a degree pattern d/sub 0//spl ges/d/sub 1//spl ges//spl middot//spl middot//spl middot//spl ges/d/sub n//spl ges/0.