Bivariate Polynomial Multiplication
نویسنده
چکیده
Inspired by the discussion in [5], we study the multiplicative complexity and the rank of the multiplication in the local algebras Rm;n = k[x; y]=(xm+1; yn+1) and Tn = k[x; y]=(xn+1; xny; : : : ; yn+1) of bivariate polynomials. We obtain the lower bounds (2 13 o(1)) dimRm;n and (2 12 o(1)) dim Tn for the multiplicative complexity of the multiplication in Rm;n and Tn, respectively. On the other hand, we derive the upper bounds 3 dim Tn 2n 2 and 3 dimRm;n m n 3 for the rank of the multiplication in Tn and Rm;n, respectively, provided that the ground field k admits “fast” univariate polynomial multiplication mod xN 1. Our results are also applicable to arbitrary finite dimensional algebras of truncated bivariate polynomials k[x; y]=I , where the ideal I = (xd0+1; xd1+1y; : : : ; xdn+1yn; yn+1) is described by a degree pattern d0 d1 dn 0.
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