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arXiv:1309.2151 [math.CA]AbstractReferencesReviewsResources

Regularity of roots of polynomials

Adam Parusinski, Armin Rainer

Published 2013-09-09, updated 2015-02-11Version 2

We show that smooth curves of monic complex polynomials $P_a (Z)=Z^n+\sum_{j=1}^n a_j Z^{n-j}$, $a_j : I \to \mathbb C$ with $I \subset \mathbb R$ a compact interval, have absolutely continuous roots in a uniform way. More precisely, there exists a positive integer $k$ and a rational number $p >1$, both depending only on the degree $n$, such that if $a_j \in C^{k}$ then any continuous choice of roots of $P_a$ is absolutely continuous with derivatives in $L^q$ for all $1 \le q < p$, in a uniform way with respect to $\max_j\|a_j\|_{C^k}$. The uniformity allows us to deduce also a multiparameter version of this result. The proof is based on formulas for the roots of the universal polynomial $P_a$ in terms of its coefficients $a_j$ which we derive using resolution of singularities. For cubic polynomials we compute the formulas as well as bounds for $k$ and $p$ explicitly.

Comments: 32 pages, 2 figures; minor changes; accepted for publication in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)
Categories: math.CA, math.AG
Subjects: 26C10, 26A46, 30C15, 32S45
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