The Emergence of 4-Cycles in Polynomial Maps over the Extended Integers
Andrew Best
The Ohio State University
Patrick Dynes
Clemson University
Steven J. Miller
Williams College
Jasmine Powell
University of Michigan
Benjamin Weiss
University of Maine, Orono
Abstract
Let $f(x) \in \Z[x]$; for each integer $\alpha$ it is interesting to consider the number of iterates $n_{\alpha}$, if possible, needed to satisfy $f^{n_{\alpha}}(\alpha) = \alpha$. The sets $\{\alpha, f(\alpha), \ldots, f^{n_{\alpha} - 1}(\alpha)\}$ generated by the iterates of $f$ are called cycles. For $\Z[x]$ it is known that cycles of length 1 and 2 occur, and no others. While much is known for extensions to number fields, we concentrate on extending $\Z$ by adjoining reciprocals of primes. Let $\Z[1/p_1, \ldots, 1/p_n]$ denote $\Z$ extended by adding in the reciprocals of the $n$ primes $p_1, \ldots, p_n$ and all their products and powers with each other and the elements of $\Z$.
Interestingly, cycles of length 4, called 4-cycles, emerge under the appropriate conditions for polynomials in $\Z\left[1/p_1, \ldots, 1/p_n\right][x]$. The problem of finding criteria under which 4-cycles emerge is equivalent to determining how often a sum of four terms is zero, where the terms are $\pm 1$ times a product of elements from the list of $n$ primes. We investigate conditions on sets of primes under which 4-cycles emerge. We characterize when 4-cycles emerge if the set has one or two primes, and (assuming a generalization of the ABC conjecture) find conditions on sets of primes guaranteed not to cause 4-cycles to emerge.
