Linear Diophantine equations and conjugator length in 2-step nilpotent groups

We establish upper bounds on the lengths of minimal conjugators in 2-step nilpotent groups. These bounds exploit the existence of small integral solutions to systems of linear Diophantine equations. We prove that in some cases these bounds are sharp. This enables us to construct a family of finitely generated 2-step nilpotent groups $(G_m{)}_{m\in \mathbb{N}}$ such that the conjugator length function of $G_m$ grows like a polynomial of degree $m+1$.