Michael E. Zieve:
Cycles of polynomial mappings,
Ph.D. thesis, Berkeley, 1996.

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)=x2g(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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