Specifying attracting cycles for Newton maps of polynomials
| Autor: | Jared T. Collins, James T. Campbell |
|---|---|
| Rok vydání: | 2013 |
| Předmět: |
Polynomial
Algebra and Number Theory Degree (graph theory) Constructive proof Mathematics - Complex Variables Applied Mathematics Dynamical Systems (math.DS) Combinatorics symbols.namesake 30D05 34M03 37F10 39B12 Newton fractal FOS: Mathematics symbols Degree of a polynomial Mathematics - Dynamical Systems Complex Variables (math.CV) Newton's identities Newton's method Complex plane Analysis Mathematics |
| Zdroj: | Journal of Difference Equations and Applications. 19:1361-1379 |
| ISSN: | 1563-5120 1023-6198 |
| DOI: | 10.1080/10236198.2012.751987 |
| Popis: | 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. Comment: 18 pages, 2 figures |
| Databáze: | OpenAIRE |
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