Let R be a ring and let g(X) be a univariate polynomial over R. We study the cycles of g in R, namely the n-typles of distinct elements x1, ..., xn in R such that g(x1)=x2, g(x2)=x3, ..., g(xn-1)=xn, and g(xn)=x1. We show that in many situations there are severe constraints on the integers n which occur as cycle lengths. Our most difficult results address discrete valuation rings R. In this case it is not hard to control the prime-to-p part of the cycle length, where p is the characteristic of the residue field of R; we give sharp bounds on the power of p occurring in the cycle length.
We also study various further properties of cycles of polynomial mappings.
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