The Root Extraction Problem for Generic Braids

Abstract

We show that, generically, finding the k-th root of a braid is very fast. More precisely, we provide an algorithm which, given a braid x on n strands and canonical length l, and an integer k > 1 , computes a k-th root of x, if it exists, or guarantees that such a root does not exist. The generic-case complexity of this algorithm is O ( l ( l + n ) n 3 log n ) . The non-generic cases are treated using a previously known algorithm by Sang-Jin Lee. This algorithm uses the fact that the ultra summit set of a braid is, generically, very small and symmetric (through conjugation by the Garside element Δ ), consisting of either a single orbit conjugated to itself by Δ or two orbits conjugated to each other by Δ .