Perturbations of Complex Polynomials: Regularity of the Roots

Abstract

We study the regularity of roots of complex polynomials $P\left(t\right)\left(Z\right)=Z^n+{\sum }_{j=1}^n{a}_j\left(t\right)Z^{n-j}$ depending on a real parameter t. We first give an overview of the known results including in particular the hyperbolic case (Rellich’s and Bronshtein’s Theorems).

Then we show, in the general case, that if the coefficients a˙j(t) are sufficiently regular ($C^k$ for $k=k\left(n\right)$ large) then any continuous choice of roots is locally absolutely continuous. This solves a problem that was open for more then a decade and implies that some systems of pseudodifferential equations are solvable. Our main tool is the resolution of singularities.

This is joint work with Armin Rainer from Vienna.