Narkiewicz: Polynomial cycles in certain rings of rationals

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Władysław Narkiewicz
Polynomial cycles in certain rings of rationals
Journal de théorie des nombres de Bordeaux, 14 no. 2 (2002), p. 529-552, doi: 10.5802/jtnb.373
Article PDF | Reviews Zbl 1071.11017

Résumé - Abstract

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 $\mathbf{Z} [\frac{1}{N}]$ and shall describe polynomial cycles in the case when $N$ is either odd or twice a prime.

Bibliography

[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.
[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. Article | 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. Article | 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. Article | 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. Article | 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