Sendov’s Conjecture and the Geometry of Cubic Polynomials

Abstract

Sendov’s conjecture proposes a tight upper bound for the distance from a zero of a polynomial having roots in the unit
disk to the closest critical point. In the particular case of cubic polynomials, the Siebeck-Marden theorem provides a geometric
relation between roots and critical points. Based on this, geometric arguments are employed to prove Sendov’s conjecture for cubic
polynomials and explore its sharpness.