We consider the classical problems of interpolating a polynomial given a black box for evaluation, and of multiplying two polynomials, in the setting where the bit-lengths of the coefficients may vary widely, so-called unbalanced polynomials. Writing s for the total bit-length and D for the degree, our new algorithms have expected running time $\tilde{O}(s \log D)$, whereas previous methods for (resp.) dense or sparse arithmetic have at least $\tilde{O}(sD)$ or $\tilde{O}(s^2)$ bit complexity.

翻译：我们考虑经典的多项式插值问题（给定黑箱评估函数）与多项式乘法问题，其中系数的比特长度可能差异显著，即所谓的不平衡多项式。设s为总比特长度，D为次数，我们提出的新算法期望运行时间为$\tilde{O}(s \log D)$，而此前针对稠密或稀疏算术的方法其比特复杂度分别至少为$\tilde{O}(sD)$或$\tilde{O}(s^2)$。