Sparse polynomial representations are used in a number of algebraic manipulation systems, including Altran. This paper discusses the arithmetic operations with sparsely represented polynomials; we give particular attention to multiplication and division. We give new algorithms for multiplying two polynomials, with <i>n</i> and <i>m</i> terms, in time <i>mn</i>log<i>m</i>;, these algorithms have the property that, in the usual univariate dense case, the algorithm is bounded by <i>mn.</i> Division algorithms are discussed which run in comparable time.
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