Stability of roots of polynomials under linear combinations of derivatives

Abstract

Let $T={\alpha }_{0}I+{\alpha }_{1}D+...+{\alpha }_n{D}^n$, where $D$ is the differentiation operator and ${\alpha }_0\ne 0$, and let $f$ be a square-free polynomial with large minimum root separation. We prove that the roots of $Tf$ are close to the roots of $f$ translated by $-{\alpha }_1/{\alpha }_0$.

arXiv comment18 pages, 4 figures