Regularity of roots of polynomials

Abstract

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 ${\max }_j\Vert a_j{\Vert }_{C^k}$. 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.

32 pages, 2 figures; minor changes; accepted for publication in Ann. Sc. Norm. Super. Pisa Cl. Sci. (5); some typos corrected

References in corpus (2)

• A new proof of Bronshtein's theorem
• Lifting smooth curves over invariants for representations of compact Lie groups. II

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