Regularity of roots of polynomials

TL;DR

This paper proves that roots of smooth polynomial families depend in an absolutely continuous manner on parameters, with uniform bounds, using resolution of singularities, and provides explicit results for cubic polynomials.

Contribution

It establishes uniform absolute continuity of roots for smooth polynomial families and derives explicit bounds for cubic cases using resolution of singularities.

Findings

Roots are absolutely continuous functions of parameters under smoothness conditions.

Uniform bounds on derivatives of roots are obtained depending only on polynomial degree.

Explicit formulas and bounds are provided for cubic 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 $\max_j\|a_j\|_{C^k}$. 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.