  We prove that the roots of a definable $C^\infty$ curve of monic hyperbolic
polynomials admit a definable $C^\infty$ parameterization, where `definable'
refers to any fixed o-minimal structure on $(\mathbb R,+,\cdot)$. Moreover, we
provide sufficient conditions, in terms of the differentiability of the
coefficients and the order of contact of the roots, for the existence of $C^p$
(for $p \in \mathbb N$) arrangements of the roots in both the definable and the
non-definable case. These conditions are sharp in the definable and under an
additional assumption also in the non-definable case. In particular, we obtain
a simple proof of Bronshtein's theorem in the definable setting. We prove that
the roots of definable $C^\infty$ curves of complex polynomials can be
desingularized by means of local power substitutions $t \mapsto \pm t^N$. For a
definable continuous curve of complex polynomials we show that any continuous
choice of roots is actually locally absolutely continuous.
