Quasi-optimal arithmetic for quaternion polynomials

Fast algorithms for arithmetic on real or complex polynomials are well-known and have proven to be not only asymptotically efficient but also very practical. Based on Fast Fourier Transform, they for instance multiply two polynomials of degree up to n or multi-evaluate one at n points simultaneously within quasi-linear time O(n (.) polylog n). An extension to (and in fact the mere definition of) polynomials over fields R and C to the skew-field H of quaternions is promising but still missing. The present work proposes three approaches which in the commutative case coincide but for H turn out to differ, each one satisfying some desirable properties while lacking others. For each notion, we devise algorithms for according arithmetic; these are quasi-optimal in that their running times match lower complexity bounds up to polylogarithmic factors.