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Combinatorial cycles of a polynomial map over a commutative field. (English) Zbl 0596.12020
A map g from a field K into itself is said to have a cycle of length n if there exist n distinct elements \(x_ 1,...,x_ n\) of K such that \(g(x_ i)=x_{i+1}\) for \(1\leq i\leq n-1\) and \(g(x_ n)=x_ 1\). Let K be algebraically closed and let \(f\in K[x]\) with deg(f)\(\geq 2\). Let t be a prime number different from the characteristic of K and different from the multiplicative orders of the roots of unity u of the form \(u=f'(c)\) with \(f(c)=c\). Then it is shown that the map from K into itself induced by f has a cycle of length t. For finite fields K the action of the Frobenius automorphism on the cycles of polynomial maps is also studied.
Reviewer: H.Niederreiter

MSC:
12E05 Polynomials in general fields (irreducibility, etc.)
11T06 Polynomials over finite fields
05A99 Enumerative combinatorics
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References:
[1] Knuth, D.E, ()
[2] Pollard, J.M, A Monte-Carlo method for factorization, B.i.t., 15, 331-334, (1975) · Zbl 0312.10006
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