Cubic Self-inversive Polynomials whose roots envelope conics

TL;DR

This paper investigates a family of cubic self-inversive polynomials with roots on the unit circle, demonstrating that the triangle formed by these roots is tangent to a common conic, such as an ellipse or hyperbola, regardless of a parameter.

Contribution

It introduces a specific family of cubic self-inversive polynomials whose roots' geometric configuration envelopes conic sections, revealing a new link between polynomial roots and conic tangency.

Findings

Roots on the unit circle form triangles tangent to a common conic.

The conic can be an ellipse or hyperbola, independent of the polynomial parameter.

The geometric property holds for the entire family of polynomials studied.

Abstract

We study a one parameter family of cubic self-inversive polynomials that "envelope" conic sections in the following sense. Provided the three roots of the polynomial lie on the unit circle, when you draw the triangle connecting the roots, the sides of the triangle, or extensions thereof, will all be tangent to the same ellipse or hyperbola independent of the parameter.