Abstract
We observe that polynomial evaluation and interpolation can be performed fast over a multidimensional grid (lattice), and we apply this observation in order to devise a simple algorithm for multivariate polynomial multiplication. Surprisingly, this simple idea enables us to improve the known algorithms for multivariate polynomial multiplication based on the forward and backward application of Kronecker's map; in particular, we decrease, by the factor log log N, the known upper bound on the arithmetic time-complexity of this computation (over any field of constants), provided that the degree d in each of the m variables is fixed, m grows to the infinity, and N = ( d + 1) m .
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