**Introduction**

Each quartic, monic polynomial of the form *p(x) = x*^{4} + a x^{2} + b x + c can be represented by a point *(a, b, c)* in parametric space. A polynomial will have a double root *r* if and only if *p(r) = 0* and *p'(r) = 0*. The set of points *(a, b, c)* which correspond to polynomials with a double root forms a boundary in parametric space between the points representing the quartics with zero, two, and four roots. (For a discussion of this phenomenon with cubic polynomials, see "Project One".) This boundary surface is known as the swallowtail. (Refer to graphic of the swallowtail at the top of this page.) The parametrization of the swallowtail is *(a, b, c) = (u, -2 u v - 4 v*^{3}, 3 v^{4} + u v^{2}).

The intent of this paper is to describe and classify the regions in parametric space defined by the swallowtail with respect to the number and position of roots of monic, quartic polynomials in real space. For this purpose, we chose several paths through parametric space to examine resulting changes to the polynomial in real space as a point moves along the chosen path.

**Contents**

Overview

Path One

Path Two

Path Three

Path Four

Points on a Tangent Plane