ISSN:
1420-8903

Keywords:
Primary 10A50, 39A10, 10L20
;
Secondary 39A99

Source:
Springer Online Journal Archives 1860-2000

Topics:
Mathematics

Notes:
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, $$ - \frac{{2^{k - 1} }}{{k!}}〈 f\left[ {x_1 ,... + x_{i + k} } \right]〈 \frac{{x_{i + k + 1} - x_{i + k} + 2^{k - 1} }}{{k!}},$$ 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 $$f(x) = \sum\limits_{j = 0}^n {a_j x^j } ,n \geqq 2$$ . 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. Ap 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 mostk with integer coefficients and satisfyingf(x j )=x j+1 forj=1, 2, ...,n−1. Definep 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 ap k -sequence of lengthn. THEOREM. pn−2(n)≦(n!)(n−2)!/2.

Type of Medium:
Electronic Resource

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