Classification of the Real Roots of the Quartic Equation and their Pythagorean Tunes

Abstract

Presented is a very detailed two-tier analysis of the location of the real roots of the general quartic equation $$x^4 + a x^3 + b x^2 + c x + d = 0$$\n \n \n x\n 4\n \n +\n a\n \n x\n 3\n \n +\n b\n \n x\n 2\n \n +\n c\n x\n +\n d\n =\n 0\n \n with real coefficients and the classification of the roots in terms of a, b, c, and d, without using any numerical approximations. Associated with the general quartic, there is a number of subsidiary quadratic equations (resolvent quadratic equations) whose roots allow this systematization as well as the determination of the bounds of the individual roots of the quartic. In many cases the root isolation intervals are found. The second tier of the analysis uses two subsidiary cubic equations (auxiliary cubic equations) and solving these, together with some of the resolvent quadratic equations, allows the full classification of the roots of the general quartic and also the determination of the isolation interval of each root. These isolation intervals involve the stationary points of the quartic (among others) and, by solving some of the resolvent quadratic equations, the isolation intervals of the stationary points of the quartic are also determined. The presented classification of the roots of the quartic equation is particularly useful in situations in which the equation stems from a model the coefficients of which are (functions of) the model parameters and solving cubic equations, let alone using the explicit quartic formulæ\xa0, is a daunting task. The only benefit in such cases would be to gain insight into the location of the roots and the proposed method provides this. Each possible case has been carefully studied and illustrated with a detailed figure containing a description of its specific characteristics, analysis based on solving cubic equations and analysis based on solving quadratic equations only. As the analysis of the roots of the quartic equation is done by studying the intersection points of the “sub-quartic” $$x^4 + ax^3 + bx^2$$\n \n \n x\n 4\n \n +\n a\n \n x\n 3\n \n +\n b\n \n x\n 2\n \n \n with a set of suitable parallel lines, a beautiful Pythagorean analogy can be found between these intersection points and the set of parallel lines on one hand and the musical notes and the staves representing different musical pitches on the other: each particular case of the quartic equation has its own short tune.