Combinatorial cycles of a polynomial map over a commutative field

@article{Chass1986CombinatorialCO,
  title={Combinatorial cycles of a polynomial map over a commutative field},
  author={G. Chass{\'e}},
  journal={Discret. Math.},
  year={1986},
  volume={61},
  pages={21-26}
}
  • G. Chassé
  • Published 1986
  • Computer Science, Mathematics
  • Discret. Math.
Abstract Let K be a commutative field and f : K → K a polynomial map. We show that, if the degree of f as a polynomial is greater than 1, then the cycle length of f , extended to an algebraic closure K of K is not bounded. That is to say that, for each positive integer N , one can find an integer n , n ⩾ N such that there exist n different elements x 1 ,…, x n of K with the property: f ( x i ) = x i +1 for i , 1⩽ i ⩽ n − 1 and f ( x n ) = x 1 . 
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References

SHOWING 1-2 OF 2 REFERENCES
A monte carlo method for factorization
We describe briefly a novel factorization method involving probabilistic ideas.
The Art of Computer Programming
  • Vol. 2 Seminumerical Algorithms, 2nd ed.
  • 1981