Let Pn be the complex vector space of all polynomials of degree at most n. We give several characterizations of the linear operators T:Pn→Pn for which there exists a constant C > 0 such that for all nonconstant f∈Pn there exist a root u of f and a root v of Tf with |u−v|≤C. We prove that such perturbations leave the degree unchanged and, for a suitable pairing of the roots of f and Tf, the roots are never displaced by more than a uniform constant independent on f. We show that such "good" operators T are exactly the invertible elements of the commutative algebra generated by the differentiation operator. We provide upper bounds in terms of T for the relevant constants.
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Ćurgus, Branko and Mascioni, Vania, "Perturbations of Roots under Linear Transformations of Polynomials" (2007). Mathematics. Paper 68.