Power Series Composition in Near-Linear Time

We present an algebraic algorithm that computes the composition of two power series in softly linear time complexity. The previous best algorithms are <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{O}(n^{1+o(1)})$</tex> non-alzebraic algorithm by Kedlaya and Umans (FOCS 2008) and an <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{O}(n^{1.43})$</tex> 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).