Smooth roots of hyperbolic polynomials with definable coefficients

Abstract

We prove that the roots of a definable C ^∞ curve of monic hyperbolic polynomials admit a definable C ^∞ parameterization, where ‘definable’ refers to any fixed o-minimal structure on (ℝ,+, ·). 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 ∈ 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 ^∞ curves of complex polynomials can be desingularized by means of local power substitutions t ↦ ±t ^N. For a definable continuous curve of complex polynomials we show that any continuous choice of roots is actually locally absolutely continuous.