Smooth roots of hyperbolic polynomials with definable coefficients

TL;DR

This paper proves that roots of definable smooth hyperbolic polynomial curves can be smoothly parameterized and provides conditions for roots to be differentiable, extending classical results to the o-minimal setting.

Contribution

It establishes the existence of smooth root parameterizations for definable hyperbolic polynomials and sharp conditions for differentiability, including a new proof of Bronshtein's theorem.

Findings

Roots admit definable $C^ abla$ parameterizations.

Conditions for $C^p$ arrangements are sharp.

Roots can be desingularized via power substitutions.

Abstract

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.