Let $ {\mbox{${\mathbb{K}}$}}$ be a field. We assume that we have at our disposal a fast algorithm for multiplications in $ {\mbox{${\mathbb{K}}$}}[x_1]$ . Let $ M(d)$ be an upper bound for the number of operations in $ {\mathbb{K}}$ required for computing the product of two polynomials $ {\mbox{${\mathbb{K}}$}}[x_1]$ with degree at most $ d$ .

We would like to derive a fast algorithm for multiplying bivariate polynomials in $ {\mbox{${\mathbb{K}}$}}[x_1,x_2]$ . To do so, we consider the following transformation.

Let $ {\alpha}_2 > 0$ be a positive integer. Given a bivariate polynomial $ f$ we replace every monomial $ x_1^{e_1} x_2^{e_2}$ of $ f$ by $ x_1^{e_1 + {\alpha}_2 e_2}$ obtaining a univariate polynomial $ u_f$ . Let $ d_1$ be the degree of $ f$ w.r.t. $ x_1$ and $ d_2$ its degree w.r.t. $ x_2$ ,

Marc Moreno Maza