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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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Online Encyclopedia of Integer Sequences:

Number of irreducible factors of x^(2n+1) - 1 over GF(2).

References:

[1] Knuth, D. E., (The Art of Computer Programming, Vol. 2 Seminumerical Algorithms (1981), Addison-Wesley: Addison-Wesley Reading)
[2] Pollard, J. M., A Monte-Carlo method for factorization, B.I.T., 15, 331-334 (1975) · Zbl 0312.10006
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