David desJardins and Michael E. Zieve:
Polynomial mappings mod pn,
preprint, 1994.

(Prior to publication, this paper should be cited as arXiv:math/0103046.)

We study the map  cg(c)  on  Z / pnZ,  where  g(X)  is a polynomial with integer coefficients and  p  is an odd prime. If  g  permutes  Z / pnZ,  we show how the cycle structure of this permutation is determined by the cycle structure of the corresponding permutation of  Z / p3Z.  More generally, we prove similar results about subsets of  Z / pnZ  which are permuted by  g.  For instance, if  g  transitively permutes a subset  S  of  Z / pnZ  with  n>2,  and if reduction mod  pn-1  is a  p-to-one map on  S,  then  g  transitively permutes the  pk #S  elements of  Z / pn+kZ  which reduce mod  pn  to elements of  S


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