Power Series Composition in Near-Linear Time

TL;DR

The paper addresses the problem of efficiently computing the composition of two power series, $f(g(x)) \bmod x^n$, over an arbitrary commutative ring. It introduces a near-linear-time, algebraic approach based on Graeffe iteration applied to a special bivariate rational series $P(x)/Q(x,y)$ with $Q(x,y)=1- y g(x)$ and a halving/doubling recursion across $\log n$ steps, combined with the transposition principle to relate power projection to composition. The main result is a complexity bound of $O(\mathsf{M}(n)\log m + \mathsf{M}(m))$ arithmetic operations (with $\deg f < m$ and $\deg g < n$), extending beyond field-specific limits and enabling general-ring applicability. The authors also provide a direct, self-contained derivation of the composition algorithm in addition to the transposed, power-projection-based route, highlighting the method’s potential impact on modular composition and related power-series computations in computer algebra.

Abstract

We present an algebraic algorithm that computes the composition of two power series in softly linear 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).

Theorems & Definitions (9)

• Theorem 1
• Theorem 2: kung1974computing
• Theorem 3: ModernComputerAlgebra
• Lemma 4: brent1978fast
• Theorem 5: BCS10AlgComp
• Definition 1: Power Projection
• Theorem 6
• proof
• proof : Proof of Theorem \ref{['thm:main']}