A note on the fast power series' exponential

TL;DR

This paper presents efficient algorithms for computing the exponential of a complex power series and raising a power series to a constant power, optimizing the number of binary operations based on polynomial multiplication complexity.

Contribution

It introduces new complexity bounds for exponential and power operations on power series, improving upon previous methods using FFT-based polynomial multiplication.

Findings

Exponential of a power series can be computed with approximately 1.92 times the polynomial multiplication complexity.

Raising a power series to a constant power requires about 3.375 times the polynomial multiplication complexity.

The methods optimize the number of binary operations needed for these computations.

Abstract

It is shown that the exponential of a complex power series up to order n can be implemented via (23/12+o(1))M(n) binary arithmetic operations over complex field, where M(n) stands for the (smoothed) complexity of multiplication of polynomials of degree <n in FFT-model. Yet, it is shown how to raise a power series to a constant power with the complexity (27/8+o(1))M(n).