Complex Ratios of Cubic Polynomials

Details Complex Ratios of Cubic Polynomials Complex Ratios of Cubic Polynomials

2007-06-03

arxiv.org/abs/0706.0346v1

International Journal of Pure and Applied Mathematics, vol. 33, No. 1 (2006), 49-62

Mathematics

Complex Variables

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Complex Ratios of Cubic Polynomials does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Complex Ratios of Cubic Polynomials, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Complex Ratios of Cubic Polynomials will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-726512