Polynomial Equations and Circulant Matrices

1. INTRODUCTION. There is something fascinating about procedures for solving low degree polynomial equations. On one hand, we all know that while general solutions (using radicals) are impossible beyond the fourth degree, they have been found for quadratics, cubics, and quartics. On the other hand, the standard solutions for the the cubic and quartic are complicated, and the methods seem ad hoc. How is a person supposed to remember them? It just seems that there ought to be a simple, memorable, unified method for all equations through degree four. Approaches to unification have been around almost as long as the solutions themselves. In 1545, Cardano published solutions to both the cubic and quartic, attributing the former to Tartaglia and the latter to Ferrari. After subsequent work failed to solve equations of higher degree, Lagrange undertook an analysis in 1770 to explain why the methods for cubics and quartics are successful. From that time right down to the present, efforts have persisted to illuminate the solutions of cubic and quartic equations ; see [21]. In this paper we present a unified approach based on circulant matrices. The idea is to construct a circulant matrix with a specified characteristic polynomial. The roots of the polynomial thus become eigenvalues, which are trivially found for circulant matrices. This circulant matrix approach provides a beautiful unity to the solutions of cubic and quartic equations, in a form that is easy to remember. It also reveals other interesting insights and connections between matrices and polynomials, as well as cameo roles for interpolation theory and the discrete Fourier transform. We begin with a brief review of circulants, and then show how circulants can be used to find the zeroes of low degree polynomials. Succeeding sections explore how the circulant method is related to other approaches, present additional applications of circulants in the study of polynomial roots, and discuss generalizations using other classes of matrices.