Specifying attracting cycles for Newton maps of polynomials

TL;DR

This paper proves that for any set of n distinct points, there exists a polynomial of degree at most n+1 whose Newton map has these points as a super-attracting cycle, improving previous degree bounds.

Contribution

The authors establish a tighter degree bound for polynomials with prescribed super-attracting cycles in their Newton maps, and provide a constructive proof of existence.

Findings

Existence of polynomials of degree at most n+1 with specified super-attracting cycles

Improvement over previous degree bounds of 2n

Constructive proof of cycle existence for given lengths and degrees

Abstract

We show that for any set of n distinct points in the complex plane, there exists a polynomial p of degree at most n+1 so that the corresponding Newton map, or even the relaxed Newton map, for p has the given points as a super-attracting cycle. This improves the result due to Plaza and Romero (2011), which shows how to find such a polynomial of degree 2n. Moreover we show that in general one cannot improve upon degree n+1. Our methods allow us to give a simple, constructive proof of the known result that for each cycle length n at least 2 and degree d at least 3, there exists a polynomial of degree d whose Newton map has a super-attracting cycle of length n.