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：本文研究了在系数位长变化较大（即所谓的不平衡多项式）的设定下，给定评估黑盒的多项式插值经典问题以及两个多项式相乘的经典问题。记总位长为 s、次数为 D，我们提出的新算法期望运行时间为 $\tilde{O}(s \log D)$，而先前针对稠密或稀疏算术的方法分别至少具有 $\tilde{O}(sD)$ 或 $\tilde{O}(s^2)$ 的比特复杂度。