The Geometry of Cubic Polynomials

Given a complex cubic polynomial p(z) = (z − 1)(z − r1)(z − r2) with |r1| = 1 = |r2|, where are the critical points? Marden’s Theorem tells us that the critical points are the foci of the Steiner ellipse of 41r1r2. In this paper we further explore the structure of these critical points. If we let Tγ be the circle of diameter γ passing through 1 and 1 − γ, then there are α, β ∈ [0, 2] such that the critical points of p lie on the circles Tα and Tβ respectively. We show that Tβ is the inversion of Tα over T1, from which many nice geometric consequences can be drawn. For example, (1) there is a “desert” in the unit disk, the open disk {z ∈ C : |z − 2 3 | < 1 3 }, in which critical points cannot occur, and (2) a critical point of such a polynomial almost always determines the polynomial uniquely.