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


12E05 Polynomials in general fields (irreducibility, etc.)
11T06 Polynomials over finite fields
05A99 Enumerative combinatorics
Full Text: DOI

Online Encyclopedia of Integer Sequences:

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


[1] Knuth, D. E., (The Art of Computer Programming, Vol. 2 Seminumerical Algorithms (1981), Addison-Wesley: Addison-Wesley Reading) · Zbl 0477.65002
[2] Pollard, J. M., A Monte-Carlo method for factorization, B.I.T., 15, 331-334 (1975) · Zbl 0312.10006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.