Sparse polynomial multiplication is a fundamental problem in computer algebra and the theory of computation, and the development of a quasi-linear time output-sensitive multiplication algorithm has been posed as an open challenge. In this paper, a counterexample is provided to a previously claimed solution to this open problem for integer coefficients. By employing the existing quasi-linear modular-black-box interpolation algorithm, we are able to provide an algorithm with quasi-linear bit complexity for the integer coefficients setting. Furthermore, in the case of coefficients over a finite field, we obtain an algorithm whose bit complexity is linear in the number of terms, the logarithm of the degree, and the logarithm of the size of the finite field.

翻译：稀疏多项式乘法是计算机代数与计算理论中的基本问题，实现拟线性时间输出敏感乘法算法长期被视作一项开放性挑战。本文针对整数系数情形，为先前声称解决该开放问题的方案提供了一个反例。通过采用现有拟线性模黑箱插值算法，我们能够在整数系数场景下给出具有拟线性比特复杂度的算法。此外，当系数取自有限域时，我们获得的算法其比特复杂度关于项数、度数的对数以及有限域大小的对数均呈线性。