Hyperbolic components and cubic polynomials

Triangles, Ellipses, and Cubic Polynomials

1. INTRODUCTION. Discussions that led to this paper began during an electronic version of the Secondary School Teachers Program, a part of the Park City Mathematics Institute (PCMI), offered at the University of Cincinnati in the summer of 2006 for local high school mathematics teachers. The authors were working with the teachers on daily problem sets provided by PCMI; complex numbers and polyn...

Geometry of Cubic Polynomials

Imagine a sphere with its equator inscribed in an equilateral triangle. This Saturn-like figure will help us understand from where Cardano’s formula for finding the roots of a cubic polynomial p(z) comes. It will also help us find a new proof of Marden’s theorem, the surprising result that the roots of the derivative p(z) are the foci of the ellipse inscribed in and tangent to the midpoints of ...

Reducible cubic CNS polynomials

The concept of a canonical number system can be regarded as a natural generalization of decimal representations of rational integers to elements of residue class rings of polynomial rings. Generators of canonical number systems are CNS polynomials which are known in the linear and quadratic cases, but whose complete description is still open. In the present note reducible CNS polynomials are tr...

Hyperbolic Polynomials and Spectral Order

Hyperbolic Polynomials and Convex Analysis

A homogeneous real polynomial p is hyperbolic with respect to a given vector d if the univariate polynomial t → p(x − td) has all real roots for all vectors x. Motivated by partial differential equations, Gårding proved in 1951 that the largest such root is a convex function of x, and showed various ways of constructing new hyperbolic polynomials. We present a powerful new such construction, an...