Geodesic
On some issues concerning polynomial cycles
Communications in Mathematics, Tome 21 (2013) no. 2, pp. 129-135 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
Classification : 11R09, 13F05, 37P35
Keywords: polynomial cycles; discrete valuation domains; Dedekind rings 
@article{COMIM_2013_21_2_a2,
     author = {Pezda, Tadeusz},
     title = {On some issues concerning polynomial cycles},
     journal = {Communications in Mathematics},
     pages = {129--135},
     year = {2013},
     volume = {21},
     number = {2},
     mrnumber = {3159285},
     zbl = {06296533},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/COMIM_2013_21_2_a2/}
}
TY  - JOUR
AU  - Pezda, Tadeusz
TI  - On some issues concerning polynomial cycles
JO  - Communications in Mathematics
PY  - 2013
SP  - 129
EP  - 135
VL  - 21
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/COMIM_2013_21_2_a2/
LA  - en
ID  - COMIM_2013_21_2_a2
ER  - 
%0 Journal Article
%A Pezda, Tadeusz
%T On some issues concerning polynomial cycles
%J Communications in Mathematics
%D 2013
%P 129-135
%V 21
%N 2
%U http://geodesic.mathdoc.fr/item/COMIM_2013_21_2_a2/
%G en
%F COMIM_2013_21_2_a2
Pezda, Tadeusz. On some issues concerning polynomial cycles. Communications in Mathematics, Tome 21 (2013) no. 2, pp. 129-135. http://geodesic.mathdoc.fr/item/COMIM_2013_21_2_a2/

[1] Narkiewicz, W.: Polynomial Mappings, Lecture Notes in Mathematics, vol. 1600. 1995, Springer-Verlag, Berlin. | MR

[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 | MR | Zbl

[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 | MR | Zbl

[4] Silverman, J.H.: The Arithmetic of Dynamical Systems, Graduate Texts in Mathematics, No. 241. 2007, Springer-Verlag. | DOI | MR

[5] Zieve, M.: Cycles of Polynomial Mappings. PhD thesis, 1996, University of California at Berkeley. | MR