The Joukowsky Map Reveals the Cubic Equation

Abstract

Two canonical polynomials generate all cubics, via linear transformations of the polynomial map and the parameter: the cubic power function, with coincident critical points, and the third Chebyshev polynomial of the first kind, with two distinct critical points. Computing the roots of any cubic boils down to inverting these fundamental maps. In the more general case of distinct critical points, we show that the roots admit a startlingly simple expression in terms of a Joukowsky map and its inverse. Marden's theorem comes as a straightforward consequence, because the roots are the images, under a Joukowsky map, of the vertices of an equilateral triangle. The first time you come across the closed-form solution to the cubic equation [23], it looks intimidating, with square roots within cube roots and terms coming out of the blue, without apparent geometric meaning. Our goal is to change this perception, by reexamining known results and endowing them with an intuitive interpretation that leads to more compact expressions. To get this new insight, we analyze cubic polynomials as complex transformations, appealing to geometric intuition as exemplified by Needham in his beautifully illustrated book [18]. 1. CLASSIFYING CUBIC MAPS. The first question to answer is: How many essentially different cubic maps are there? We consider only true cubics, hence ruling out quadratic or linear maps. The answer arises naturally from a key feature of complex maps, namely their critical points z c , where the conformality of the otherwise conformal mapping breaks down. For a polynomial, critical points correspond to the points z c where the derivative vanishes. In particular, for a cubic c(z), the equation c (z c) = 0 is quadratic, furnishing only two cases: coincident or distinct critical points. Thus, we split the set of cubics into two subsets, depending on their number N = 1, 2 of distinct critical points. Formally speaking, we define two equivalence classes, whereby two cubics are equivalent if they share the same N. This case-by-case analysis will provide a deeper insight and prove simpler than the traditional approach of categorizing cubics according to the nature of their roots [2]. Next, we check that each class is generated by its respective canonical representative ζ(z), via linear transformations of ζ(z) and the parameter z. In the complex plane C, these transformations have a clear geometric meaning, since they amount to direct similarities [18], that is, combinations of uniform dilations, translations and rotations.