Stability of roots of polynomials under linear combinations of derivatives

TL;DR

This paper proves that applying a specific linear differential operator to a polynomial with well-separated roots results in roots that are close to the original roots shifted by a fixed amount, demonstrating stability under such transformations.

Contribution

It establishes the stability of polynomial roots under linear differential operators with constant coefficients, extending understanding of root behavior under differential transformations.

Findings

Roots of Tf are close to roots of f shifted by -α₁/α₀.

Stability holds for square-free polynomials with large root separation.

Results apply to a class of linear differential operators.

Abstract

Let $T=\alpha_0 I + \alpha_1 D + ...+\alpha_n D^n$, where $D$ is the differentiation operator and $\alpha_0\not= 0$, 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 $-\alpha_1/\alpha_0$.