The Geometry of Cubic Polynomials

Abstract

SummaryWe study the critical points of a complex cubic polynomial, normalized to have the form p(z) = (z - 1)(z - r1)(z - r2) with |r1| = 1 |r2|. If Tγ denotes the circle of diameter passing through 1 and 1 - γ, then there are α, β ∈ [0, 2] such that one critical point of p lies on Tα and the other on Tβ. We show that Tβ is the inversion of Tα over T1, from which many geometric consequences can be drawn. For example, (1) a critical point of such a polynomial almost always determines the polynomial uniquely, and (2) there is a “desert” in the unit disk, the open disk {z ϵ 핔 : |z - ⅔| < ⅓}, in which critical points cannot occur.