Polynomial cycles in certain rings of rationals. (English) Zbl 1071.11017
A finite sequence \((x_0; x_1; \ldots; x_n)\) in an integral domain \(R\) is called a polynomial cycle of length \(n\) if \(x_0; x_1; \ldots; x_{n-1}\) are distinct; \(x_n = x_0\); and there exists a polynomial \(f \in R [X]\) such that \(f(x_i) = x_{i+1}\) for \(i = 0; 1; \ldots; n-1\). For a square-free positive integer \(N\); let \(C(N)\) be the set of all lengths of polynomial cycles in \(\mathbb Z [1/N]\). The author explicitly determines \(C(N)\) if \(N\) is odd; \(N = 2\) or \(N = 2p\) for an odd prime \(p\). The investigations are based on the methods and the general finiteness results derived in [F. Halter-Koch and W. Narkiewicz, “Scarcity of finite polynomial orbits”, Publ. Math., Debrecen 56, 405-414 (2000; Zbl 0961.11005)].
Reviewer: Franz Halter-Koch (Graz)
MSC:
| 11C08 | Polynomials in number theory |
| 11D99 | Diophantine equations |
| 11R09 | Polynomials (irreducibility, etc.) |
Citations:
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