Note on the local growth of iterated polynomials

John W. Layman, Bruce M. Landman

Research output: Contribution to journalArticle

1 Citation (Scopus)

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 language English (US) 150-156 7 Aequationes Mathematicae 27 1 https://doi.org/10.1007/BF02192665 Published - Mar 1 1984 Yes

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Polynomials
Order of a polynomial
Polynomial
Integer
Difference equations
Monotonic increasing sequence
Nonlinear Difference Equations
Coefficient
Iterate
Difference equation
Increment
Consecutive
Quotient
Strictly
Upper bound
Denote
First-order
Arbitrary
Theorem
Generalization

Keywords

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

ASJC Scopus subject areas

• Mathematics(all)

Cite this

Note on the local growth of iterated polynomials. / Layman, John W.; Landman, Bruce M.

In: Aequationes Mathematicae, Vol. 27, No. 1, 01.03.1984, p. 150-156.

Research output: Contribution to journalArticle

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AB - 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.

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