### Abstract

The local behavior of the iterates of a real polynomial is investigated. The fundamental result may be stated as follows: THEOREM. Let x_{i}, for i=1, 2, ..., n+2, be defined recursively by x_{i+1}=f(x_{i}), where x_{1} is an arbitrary real number and f is a polynomial of degree n. Let x_{i+1}-x_{i}≧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[x_{i}, ..., x_{i+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 {x_{i}}, i = 1, 2, 3, ..., be the solution of the nonlinear first order difference equation x_{i+1}=f(x_{i}) where x_{1} is an arbitrarily assigned real number and f is the polynomial {Mathematical expression}. Let δ be positive with δ^{n-1}=|2^{n-1}/n!a_{n}|. Then, if n is even and a_{n}<0, there do not exist n + 1 consecutive increments Δx_{i}=x_{i+1}-x_{i} in the solution {x_{i}} with Δx_{i}≧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 p_{k}-sequence of length n is defined to be a strictly increasing sequence of positive integers {x_{1}, ..., x_{n}} for which there exists a polynomial of degree at most k with integer coefficients and satisfying f(x_{j})=x_{j+1} for j=1, 2, ..., n-1. Define p_{k}(n) to be the least positive integer such that if {1, 2, ..., p_{k}(n)} is partitioned into two sets, then one of the two sets must contain a p_{k}-sequence of length n. THEOREM. p_{n-2}(n)≦(n!)^{(n-2)!/2}.

Original language | English (US) |
---|---|

Pages (from-to) | 150-156 |

Number of pages | 7 |

Journal | Aequationes Mathematicae |

Volume | 27 |

Issue number | 1 |

DOIs | |

State | Published - Mar 1 1984 |

Externally published | Yes |

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### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Aequationes Mathematicae*,

*27*(1), 150-156. https://doi.org/10.1007/BF02192665

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

Research output: Contribution to journal › Article

*Aequationes Mathematicae*, vol. 27, no. 1, pp. 150-156. https://doi.org/10.1007/BF02192665

}

TY - JOUR

T1 - Note on the local growth of iterated polynomials

AU - Layman, John W.

AU - Landman, Bruce M.

PY - 1984/3/1

Y1 - 1984/3/1

N2 - 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.

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.

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

UR - http://www.scopus.com/inward/record.url?scp=34250141586&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34250141586&partnerID=8YFLogxK

U2 - 10.1007/BF02192665

DO - 10.1007/BF02192665

M3 - Article

VL - 27

SP - 150

EP - 156

JO - Aequationes Mathematicae

JF - Aequationes Mathematicae

SN - 0001-9054

IS - 1

ER -