Note on the local growth of iterated polynomials

John W. Layman, Bruce Landman

Research output: Contribution to journalArticle

1 Scopus citations

Abstract

The local behavior of the iterates of a real polynomial is investigated. The fundamental result may be stated as follows: THEOREM. Let xi, for i=1, 2, ..., n+2, be defined recursively by xi+1=f(xi), where x1 is an arbitrary real number and f is a polynomial of degree n. Let xi+1-xi≧1 for i=1, ..., n + 1. Then for all i, 1 ≦i≦n, and all k, 1≦k≦n+1-i, {Mathematical expression} where f[xi, ..., xi+k] denotes the Newton difference quotient. As a consequence of this theorem, the authors obtain information on the local behavior of the solutions of certain nonlinear difference equations. There are several cases, of which the following is typical: THEOREM. Let {xi}, i = 1, 2, 3, ..., be the solution of the nonlinear first order difference equation xi+1=f(xi) where x1 is an arbitrarily assigned real number and f is the polynomial {Mathematical expression}. Let δ be positive with δn-1=|2n-1/n!an|. Then, if n is even and an<0, there do not exist n + 1 consecutive increments Δxi=xi+1-xi in the solution {xi} with Δxi≧δ. The special case in which the iterated polynomial has integer coefficients leads to a "nice" upper bound on a generalization of the van der Waerden numbers. A pk-sequence of length n is defined to be a strictly increasing sequence of positive integers {x1, ..., xn} for which there exists a polynomial of degree at most k with integer coefficients and satisfying f(xj)=xj+1 for j=1, 2, ..., n-1. Define pk(n) to be the least positive integer such that if {1, 2, ..., pk(n)} is partitioned into two sets, then one of the two sets must contain a pk-sequence of length n. THEOREM. pn-2(n)≦(n!)(n-2)!/2.

Original languageEnglish (US)
Pages (from-to)150-156
Number of pages7
JournalAequationes Mathematicae
Volume27
Issue number1
DOIs
StatePublished - Mar 1 1984
Externally publishedYes

Keywords

  • AMS (1980) subject classification: Primary 10A50, 39A10, 10L20, Secondary 39A99

ASJC Scopus subject areas

  • Mathematics(all)
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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