Abstract
Let \({\cal P}_n\) be the complex vector space of all polynomials of degree at most n. We give several characterizations of the linear operators \(T:{\cal P}_n\rightarrow{\cal P}_n\) for which there exists a constant C > 0 such that for all nonconstant \(f\in{\cal P}_n\) there exist a root u of f and a root v of Tf with \(|u-v|\leq C\). 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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Curgus, B., Mascioni, V. Perturbations of Roots under Linear Transformations of Polynomials. Constr Approx 25, 255–277 (2007). https://doi.org/10.1007/s00365-006-0649-0
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DOI: https://doi.org/10.1007/s00365-006-0649-0