Combinatorial cycles of a polynomial map over a commutative field

✍ Scribed by Guy Chassé

• Book ID: 103055980
• Publisher: Elsevier Science
• Year: 1986
• Tongue: English
• Weight: 254 KB
• Volume: 61
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(86)90024-5

✦ Synopsis

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/~ of K is not bounded. That is to say that, for each positive integer N, one can find an integer n, n >I N such that there exist n different elements xl .... , x. of/C with the property: f(xi) = xi+l for i, 1 <~ i ~< n -1 and f(x.) = xl.