Smooth roots of hyperbolic polynomials with definable coefficients

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.

19 pages, 1 figure, minor corrections, to appear in Israel J. Math

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