Article
Full entry |
PDF
(0.4 MB)
Feedback
PDF
(0.4 MB)
Feedback
Keywords:
polynomial cycles; discrete valuation domains; Dedekind rings
polynomial cycles; discrete valuation domains; Dedekind rings
Summary:
We consider two issues concerning polynomial cycles. Namely, for a discrete valuation domain $R$ of positive characteristic (for $N\ge 1$) or for any Dedekind domain $R$ of positive characteristic (but only for $N\ge 2$), we give a closed formula for a set ${\cal CYCL}(R,N)$ of all possible cycle-lengths for polynomial mappings in $R^N$. Then we give a new property of sets ${\cal CYCL}(R,1)$, which refutes a kind of conjecture posed by W. Narkiewicz.
References:
[1] Narkiewicz, W.: Polynomial Mappings, Lecture Notes in Mathematics, vol. 1600. 1995, Springer-Verlag, Berlin. MR 1367962
[2] Pezda, T.: Cycles of polynomial mappings in several variables over rings of integers in finite extensions of the rationals. Acta Arith., 108, 2, 2003, 127-146. DOI 10.4064/aa108-2-4 | MR 1974518 | Zbl 1020.11066
[3] Pezda, T.: Cycles of polynomial mappings in several variables over rings of integers in finite extensions of the rationals, II. Monatsh. Math., 145, 2005, 321-331. DOI 10.1007/s00605-004-0290-z | MR 2162350 | Zbl 1197.37143
[4] Silverman, J.H.: The Arithmetic of Dynamical Systems, Graduate Texts in Mathematics, No. 241. 2007, Springer-Verlag. DOI 10.1007/978-0-387-69904-2_5 | MR 2316407
[5] Zieve, M.: Cycles of Polynomial Mappings. PhD thesis, 1996, University of California at Berkeley. MR 2694837
Search
Partner of
