Geometry of Quartic Polynomials

Given a quartic with four real roots (at least two distinct), those roots are the first coordinate projections of a regular tetrahedron in . That tetrahedron has a unique inscribed sphere, which projects onto an interval whose endpoints are the two roots of . Let ,, be the roots of ; then the points , , form an equilateral triangle whose vertices project on the critical points of the quartic (this is the "first derivative triangle"). A similar triangle is formed by negating the coordinates of these points ("conjugate first derivative triangle"). This application is relevant to the following so far open conjecture: there does not exist a quartic polynomial with four distinct rational roots such that , , and all have rational roots.

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DETAILS

Tetrahedron rotation: to rotate the tetrahedron independently of the polynomial, choose a rotation axis with the trackball and an angle with the slider.
Snapshot 1: tetrahedron vertices are projected on the - plane
Reference
[1] S. Northshield, "Geometry of Cubic Polynomials," Mathematics Magazine, 86, 2013 pp. 136–143.
[2] R. W. D. Nickalls, "The Quartic Equation: Alignement with an Equivalent Tetrahedron," The Mathematical Gazette, 96(535), 2012 pp. 49–55.
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