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In this article, we study the problem of multiplying two multivariate polynomials which are somewhat but not too sparse, typically like polynomials with convex supports. We design and analyze an algorithm which is based on blockwise decomposition of the input polynomials, and which performs the actual multiplication in an FFT model or some other more general so called “evaluated model”. If the input polynomials have total degrees at most , then, under mild assumptions on the coefficient ring, we show that their product can be computed with ring operations, where denotes the number of all the monomials of total degree at most .

**Authors:**

**Keywords: **sparse polynomial multiplication,
multivariate power series, evaluation-interpolation, algorithm

**A.M.S. subject classification: **68W30, 12-04, 30B10,
42-04