  Guba and Sapir asked if the simultaneous conjugacy problem was solvable in
Diagram Groups or, at least, for Thompson's group F. We give a solution to the
latter question using elementary techniques which rely purely on the
description of F as the group of piecewise linear orientation-preserving
homeomorphisms of the unit interval. The techniques we develop extend the ones
used by Brin and Squier allowing us to compute roots and centralizers as well.
Moreover, these techniques can be generalized to solve the same question in
larger groups of piecewise-linear homeomorphisms.
