Mathematics – Classical Analysis and ODEs
Scientific paper
2009-04-27
Israel J. Math., 184 (2011), No. 1, 157--182
Mathematics
Classical Analysis and ODEs
19 pages, 1 figure, minor corrections, to appear in Israel J. Math
Scientific paper
10.1007/s11856-011-0063-z
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.
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