Quasi-optimal Arithmetic for Quaternion Polynomials

Abstract. Fast algorithms for arithmetic on real or complex polynomials are wellknown 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 quasilinear 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. 1 Motivation Nearly 40 years after COOLEY and TUKEY [4], their Fast Fourier Transform (FFT) has provided numerous applications, among ...

• Type: Conference
• Year: 2003
• Where: ISAAC
• Authors: Martin Ziegler
• Keywords: Fast Fourier Transform, ISAAC 2003, Polylog N, Polynomial