We present a probabilistic algorithm to compute the product of two univariate sparse polynomials over a field with a number of bit operations that is quasi-linear in the size of the input and the output. Our algorithm works for any field of characteristic zero or larger than the degree. We mainly rely on sparse interpolation and on a new algorithm for verifying a sparse product that has also a quasi-linear time complexity. Using Kronecker substitution techniques we extend our result to the multivariate case.

中文翻译：

---

本质上最优稀疏多项式乘法

我们提出了一种概率算法来计算一个域上的两个单变量稀疏多项式的乘积，其中包含许多位操作，输入和输出的大小是准线性的。我们的算法适用于特征为零或大于度数的任何领域。我们主要依靠稀疏插值和一种新算法来验证具有准线性时间复杂度的稀疏乘积。使用 Kronecker 替换技术，我们将结果扩展到多变量情况。