A unified approach to root distribution with respect to parabolas, ellipses, and hyperbolas

Root distribution of real coefficient polynomials with respect to conic subregions in the complex plane, is determined by a standard half-plane test on a transformed polynomial. A unified approach to derive the pertinent transformation for a given conic curve by using its description in polar coordinates, is described. Specific transformations and root distribution criteria are given for a left parabola, a symmetric ellipse and a horizontal hyperbola. Examples are provided.