We present an algebraic algorithm that computes the composition of two power series in $\mathop{\tilde{\mathrm O}}(n)$ time complexity. The previous best algorithms are $\mathop{\mathrm O}(n^{1+o(1)})$ by Kedlaya and Umans (FOCS 2008) and an $\mathop{\mathrm O}(n^{1.43})$ algebraic algorithm by Neiger, Salvy, Schost and Villard (JACM 2023). Our algorithm builds upon the recent Graeffe iteration approach to manipulate rational power series introduced by Bostan and Mori (SOSA 2021).

翻译：我们提出了一种代数算法，能够在$\mathop{\tilde{\mathrm O}}(n)$时间复杂度内计算两个幂级数的复合。此前最优算法包括Kedlaya和Umans（FOCS 2008）的$\mathop{\mathrm O}(n^{1+o(1)})$算法，以及Neiger、Salvy、Schost和Villard（JACM 2023）的$\mathop{\mathrm O}(n^{1.43})$代数算法。我们的算法基于Bostan和Mori（SOSA 2021）提出的Graeffe迭代方法，用于操作有理幂级数。