Selections of bounded variation for roots of smooth polynomials

Abstract

We prove that the roots of a smooth monic polynomial with complex-valued coefficients defined on a bounded Lipschitz domain $\Omega$ in $\mathbb R^m$ admit a parameterization by functions of bounded variation uniformly with respect to the coefficients. This result is best possible in the sense that discontinuities of the roots are in general unavoidable due to monodromy. We show that the discontinuity set can be chosen to be a finite union of smooth hypersurfaces. On its complement the parameterization of the roots is of optimal Sobolev class $W^{1,p}$ for all $1 \le p < \frac{n}{n-1}$, where $n$ is the degree of the polynomial. All discontinuities are jump discontinuities. For all this we require the coefficients to be of class $C^{k-1,1}(\overline \Omega)$, where $k$ is a positive integer depending only on $n$ and $m$. The order of differentiability $k$ is not optimal. However, in the case of radicals, i.e., for the solutions of the equation $Z^r = f$, where $f$ is a complex-valued function and $r\in \mathbb R_{>0}$, we obtain optimal uniform bounds.