Complex Ratios of Cubic Polynomials

Abstract

Let $p(w)=(w-w_{1})(w-w_{2})(w-w_{3}),$with $\func{Re}w_{1}<\func{Re}w_{2}<\func{Re}w_{3}$. Assume that if the critical points of $p$ are not identical, then they cannot have equal real parts. Define the ratios $\sigma_{1}=\dfrac{z_{1}-w_{1}}{w_{2}-w_{1}}$ and $\sigma _{2}=\dfrac{z_{2}-w_{2}}{w_{3}-w_{2}}$. $(\sigma_{1},\sigma_{2})$ is called the \QTR{it}{ratio vector} of $p$. This extends the definition of ratio vectors given in earlier papers for polynomials of degree $n$ with all real roots. We then derive bounds on the real part, imaginary part, and modulus of the ratios and also some relations between the ratios. In particular, we prove that $\func{Re}\sigma_{1}\leq \func{Re}\sigma_{2}$. We also show that the ratios are real if and only if the roots of $p$ are collinear.