Let T=α 0 I+α 1 D+⋅⋅⋅+α n D n , where D is the differentiation operator and α0≠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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Ćurgus, Branko and Mascioni, Vania, "Stability of Roots of Polynomials Under Linear Combinations of Derivatives" (2010). Mathematics. Paper 66.