# Geometry of Quartic Polynomials

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.

p(x)

3

p''(x)

r

s

t

p'(x)

r,

s-t

3

s,

t-r

3

t,

r-s

3

p(x)

y

p

p'

p''

p'''