Double-depressed Euclidean-Descartes polynomials and associated cubic curves

Abstract

The aim of this paper is to continue the study \cite{c:m5} of quartic real polynomials $P$ having the same Euclidean norm as their Descartes resolvent $DR(P)$. The polynomial $P$ is given in its reduced form $P_d$ which means without a cubic term while double-depressed means the lack of the quadratic term. The class of these polynomials is found to consists in two  $1$-parameter families and the discriminant of cubic curve defined naturally by $DR(P_d)$ is computed. We consider also the double-depressed Descartes resolvent as a self-map of $\mathbb{R}^2$ and its fixed points are determined. Some special angles appear throughout this papers since the coefficients of $P_d$ are trigonometrical functions of the real parameter $t$.