On Gaussian Curvature of Graphs of Cubic Polynomials

Abstract

The Gaussian curvature of the graphs of cubic polynomials $f(z)=(z-a)(z-b)(z-c)$, with triangle of zeros $\triangle=\triangle(a,b,c)$, is analyzed. We introduce the composed Pythagorean mean $\mathtt{B}_{\triangle}$, a combination of the three classical Pythagorean means, which plays a central role in the description of the critical points of the Gaussian curvature. When $\triangle$ is not equilateral, the curvature has exactly five critical points: a global maximum at the barycenter, two local minima, and two saddle points, forming a rhombus whose vertices lie on the axes of the Siebeck-Marden ellipse of the triangle. We also initiate a comparative study of $\mathtt{B}_{\triangle}$ with the skew of the triangle by describing a family of polynomials whose zeros determine a specific family of isosceles triangles. The cubic case provides a concrete model strongly suggesting that the critical geometry of the Gaussian curvature of the graph can reveal both analytic and algebraic properties of the underlying holomorphic function.