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 △=△(a,b,c), is analyzed. We introduce the composed Pythagorean mean B△, 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 △ 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 B△ 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.
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BibTeX
@misc{cabrera2026gaussian,
title = {{On Gaussian Curvature of Graphs of Cubic Polynomials}},
author = {Baruc S. Cabrera and Manuel Cruz-López and Otto Romero-Germán},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38613},
eprint = {2609.38613},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38613},
}
RIS
TY - UNPB
TI - On Gaussian Curvature of Graphs of Cubic Polynomials
AU - Baruc S. Cabrera
AU - Manuel Cruz-López
AU - Otto Romero-Germán
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.38613
UR - https://arxiv.org/abs/2609.38613
AB - The Gaussian curvature of the graphs of cubic polynomials f(z)=(z−a)(z−b)(z−c), with triangle of zeros △=△(a,b,c), is analyzed. We introduce the composed Pythagorean mean B_△, 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 △ 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 B_△ 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.
ER -