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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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