Modular composition is the problem to compute the composition of two univariate polynomials modulo a third one. For polynomials with coefficients in a finite field, Kedlaya and Umans proved in 2008 that the theoretical complexity for performing this task could be made arbitrarily close to linear. Unfortunately, beyond its major theoretical impact, this result has not led to practically faster implementations yet. In this paper, we explore the particular case when the ground field is the field of computable complex numbers. Ultimately, when the precision becomes sufficiently large, we show that modular compositions may be performed in softly linear time.

Authors: Joris van der Hoeven, Grégoire Lecerf

Keywords: modular composition, factorization, algorithm, complexity

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Final version: Ultimate complexity for numerical algorithms