Stability of roots of polynomials under linear combinations of derivatives

Let $T=α_0 I + α_1 D + ...+α_n D^n$, where $D$ is the differentiation operator and $α_0\not= 0$, 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 $-α_1/α_0$.

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