Polynomial cycles in certain rings of rationals
Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 2, p. 529-552

It is shown that the methods established in [HKN3] can be effectively used to study polynomial cycles in certain rings. We shall consider the rings 𝐙[1 N] and shall describe polynomial cycles in the case when N is either odd or twice a prime.

On montre que la méthode développée dans [HKN3] peut être appliquée pour l’étude des cycles polynomiaux dans certains anneaux, notamment les anneaux 𝐙[1 N] pour lesquels nous décrivons les cycles polynomiaux lorsque N est impair ou le double d’un nombre premier.

@article{JTNB_2002__14_2_529_0,
     author = {Narkiewicz, W\l adys\l aw},
     title = {Polynomial cycles in certain rings of rationals},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {14},
     number = {2},
     year = {2002},
     pages = {529-552},
     zbl = {1071.11017},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2002__14_2_529_0}
}
Narkiewicz, Władysław. Polynomial cycles in certain rings of rationals. Journal de théorie des nombres de Bordeaux, Volume 14 (2002) no. 2, pp. 529-552. http://www.numdam.org/item/JTNB_2002__14_2_529_0/

[A1] L.J. Alex, Diophantine equations related to finite groups. Comm. Algebra 4 (1976), 77-100. | MR 424675 | Zbl 0324.20010

[A2] L.J. Alex, On the diophantine equation 1 + 2α = 3b5c + 2d3e5f. Math. Comp. 44 (1985), 267-278. | Zbl 0556.10008

[AF1] L.J. Alex, L.L. Foster, On the diophantine equation 1 + pa =2b + 2cpd. Rocky Mount. J. Math. 15 (1985), 739-761. | MR 813272 | Zbl 0589.10017

[AF2] L.J. Alex, L.L. Foster, On the diophantine equation 1 + x + y = z. Rocky Mount. J. Math. 22 (1992), 11-62. | MR 1159941 | Zbl 0760.11013

[Ba] C. Batut, D. Bernardi, H. Cohen, M. Olivier, User's Guide to PARI-GP, Bordeaux 1994.

[Ca] J.W.S. Cassels, On the equation ax - by = 1. Amer. J. Math. 75 (1953), 159-162. | MR 51851 | Zbl 0050.03703

[Co] J.H.E. Cohn, The Diophantine equation x2 + 3 = yn. Glasgow Math. J. 35 (1993), 203-206. | MR 1220562 | Zbl 0779.11015

[Da] M. Daberkow, C. Fieker, J. Kluners, M. Pohst, K. Roegner, M. Schornig, K. Wildanger, Kant V4. J. Symbolic Comput. 24 (1997), 267-283. | MR 1484479 | Zbl 0886.11070

[HKN1] F. Halter-Koch, W. Narkiewicz, Polynomial cycles in finitely generated domains. Monatsh. Math. 119 (1995), 275-279. | MR 1328818 | Zbl 0840.13003

[HKN2] F. Halter-Koch, W. Narkiewicz, Polynomial cycles and dynamical units. Proc. Conf. Analytic and Elementary Number Theory, 70-80, Wien, 1996. | Zbl 0882.12003

[HKN3] F. Halter-Koch, W. Narkiewicz, Scarcity of finite polynomial orbits. Publ. Math. Debrecen 56 (2000), 405-414. | MR 1765989 | Zbl 0961.11005

[Le] D.H. Lehmer, On a problem of Störmer. Illinois J. Math. 8 (1964), 57-79. | MR 158849 | Zbl 0124.27202

[Len] H.W. Lenstrajr., Euclidean number fields of large degree. Invent. Math. 38 (1977), 237-254. | MR 429826 | Zbl 0328.12007

[LN] A. Leutbecher, G. Niklasch, On cliques of exceptional units and Lenstra's construction of Euclidean fields. Number Theory, Lecture Notes in Math. 1380, 150-178, Springer, 1989. | MR 1009799 | Zbl 0703.11056

[MDT] D Z. Mo, R. TIJDEMAN, Exponential diophantine equations with four terms. Indag. Math.(N.S.) 3 (1992), 47-57. | MR 1157518 | Zbl 0765.11018

[Mo] P. Morton, Arithmetic properties of periodic points of quadratic maps, II. Acta Arith. 87 (1998), 89-102. | MR 1665198 | Zbl 1029.12002

[N1] W. Narkiewicz, Polynomial Mappings. Lecture Notes in Math. 1600, Springer, 1995. | MR 1367962 | Zbl 0829.11002

[NP] W. Narkiewicz, T. Pezda, Finite polynomial orbits in finitely generated domains. Monatsh. Math. 124 (1997), 309-316. | MR 1480362 | Zbl 0897.13024

[Pe] T. Pezda, Polynomial cycles in certain local domains. Acta Arith. 66 (1994), 11-22. | MR 1262650 | Zbl 0803.11063

[Pi] S.S. Pillai, On the equation 2x - 3y = 2X + 3Y. Bull. Calcutta Math. Soc. 37 (1945), 15-20. | Zbl 0063.06245

[Sch] H.P. Schlickewei, S-units equations over number fields. Invent. Math. 102 (1990), 95-107. | MR 1069241 | Zbl 0711.11017

[Sc] R. Scott, On the equation px - by = c and ax + by = cz. J. Number Theory 44 (1993), 153-165. | MR 1225949 | Zbl 0786.11020

[Si] W. Sierpinski, Sur une question concernant le nombre de diviseurs premiers d'un nombre naturel. Colloq. Math. 6 (1958), 209-210. | MR 104621 | Zbl 0086.03503

[Sk] C.M. Skinner, On the diophantine equation apx +bqy = c + dpz qw. J. Number Theory 35 (1990), 194-207. | MR 1057322 | Zbl 0703.11017

[St] C. Störmer, Quelques théorèmes sur l'équation de Pell x2 - Dy2 = ±1 et leurs applications. Skr. Vidensk.-selsk. (Christiania) I, Mat. Naturv. Kl. (1897), no.2, 1-48.

[TW] R. Tijdeman, L. Wang, Sums of products of powers of given prime numbers. Pacific J. Math. 132 (1988), 177-193; corr. ibidem 135 (1988), 396-398. | MR 929588 | Zbl 0606.10012

[Wa] C.T.C. Wall, A theorem on prime powers. Eureka 19 (1957), 10-11. | MR 83499

[Wg] L.X. Wang, Four terms equations. Indag. Math. 51 (1989), 355-361. | MR 1020029 | Zbl 0698.10011