Composing Power Series Over a Finite Ring in Essentially Linear Time

Fix a finite commutative ringR. Letuandvbe power series overR, withv(0) = 0. This paper presents an algorithm that computes the firstnterms of the compositionu(v), given the firstntermsofuandv, inn1 + o(1)ring operations. The algorithm is very fast in practice whenRhas small characteristic.