In the sparse polynomial multiplication problem, one is asked to multiply two sparse polynomials f and g in time that is proportional to the size of the input plus the size of the output. The polynomials are given via lists of their coefficients F and G, respectively. Cole and Hariharan (STOC 02) have given a nearly optimal algorithm when the coefficients are positive, and Arnold and Roche (ISSAC 15) devised an algorithm running in time proportional to the "structural sparsity" of the product, i.e. the set supp(F)+supp(G). The latter algorithm is particularly efficient when there not "too many cancellations" of coefficients in the product. In this work we give a clean, nearly optimal algorithm for the sparse polynomial multiplication problem.

中文翻译：

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近似最优稀疏多项式乘法

在稀疏多项式乘法问题中，要求将两个稀疏多项式 f 和 g 在时间上相乘，该时间与输入的大小加上输出的大小成正比。多项式分别通过其系数 F 和 G 的列表给出。当系数为正时，Cole 和 Hariharan (STOC 02) 给出了一个近乎最优的算法，而 Arnold 和 Roche (ISSAC 15) 设计了一种算法，其运行时间与产品的“结构稀疏性”成正比，即集合 supp(F )+支持(G)。当乘积中的系数没有“太多抵消”时，后一种算法特别有效。在这项工作中，我们为稀疏多项式乘法问题提供了一个干净的、近乎最优的算法。