Complex Ratios of Cubic Polynomials

TL;DR

This paper investigates the properties of ratios derived from cubic polynomials with complex roots, establishing bounds and relations, and characterizing when these ratios are real based on root collinearity.

Contribution

It extends the concept of ratio vectors to complex roots of cubic polynomials, providing bounds, relations, and geometric characterizations.

Findings

Reσ₁ ≤ Reσ₂ for the ratios.

Ratios are real if and only if roots are collinear.

Derived bounds on the ratios' real, imaginary parts, and modulus.

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.