edoc: No conditions. Results ordered -Date Deposited. 2024-06-24T01:38:46ZEPrintshttps://edoc.unibas.ch/images/uni-logo.jpghttps://edoc.unibas.ch/2020-09-02T09:41:59Z2020-09-07T07:28:20Zhttps://edoc.unibas.ch/id/eprint/59362This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/593622020-09-02T09:41:59ZScarcity of cycles for rational functions over a number fieldWe provide an explicit bound on the number of periodic points of a rational function defined over a number field, where the bound depends only on the number of primes of bad reduction and the degree of the function, and is linear in the degree. More generally, we show that there exists an explicit uniform bound on the number of periodic points for any rational function in a given finitely generated semigroup (under composition) of rational functions of degree at least 2. We show that under stronger assumptions the dependence on the degree of the map in the bounds can be removed. Jung Kyu CanciSolomon Vishkautsan2019-03-28T09:52:00Z2019-05-07T15:45:12Zhttps://edoc.unibas.ch/id/eprint/70004This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/700042019-03-28T09:52:00ZQuadratic maps with a periodic critical point of period 2We provide a complete classification of possible graphs of rational preperiodic points of endomorphisms of the projective line of degree 2 defined over the rationals with a rational periodic critical point of period 2, under the assumption that these maps have no periodic points of period at least 5. We explain how this extends results of Poonen on quadratic polynomials. We show that there are 13 possible graphs, and that such maps have at most 9 rational preperiodic points. We provide data related to the analogous classification of graphs of endomorphisms of degree 2 with a rational periodic critical point of period 3 or 4. Jung Kyu CanciSolomon Vishkautsan2019-03-28T09:51:59Z2019-05-07T15:41:39Zhttps://edoc.unibas.ch/id/eprint/70003This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/700032019-03-28T09:51:59ZPreperiodic points for rational functions defined over a rational function field of characteristic zeroLet k be an algebraic closed field of characteristic zero. Let K be the rational function field K = k(t). Let $\phi$ be a non–isotrivial rational function in K(z). We prove a bound for the cardinality of the set of K–rational preperiodic points for $\phi$ in terms of the number of places of bad reduction and the degree d of $\phi$. Jung Kyu Canci2019-03-28T09:51:58Z2019-05-07T15:43:38Zhttps://edoc.unibas.ch/id/eprint/70002This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/700022019-03-28T09:51:58ZPreperiodic points for rational functions defined over a global field in terms of good reductionLet $\phi$ be an endomorphism of the projective line defined over a global field K. We prove a bound for the cardinality of the set of K–rational preperiodic points for $\phi$ in terms of the number of places of bad reduction. The result is completely new in the function field case and it is an improvement of the number field case. Jung Kyu CanciLaura Paladino2019-03-28T09:51:52Z2019-05-07T15:23:59Zhttps://edoc.unibas.ch/id/eprint/69992This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/699922019-03-28T09:51:52ZGood reduction for endomorphisms of the projective line in terms of the branch locusLet K be a number field and v a non archimedean valuation on K. We say that an endomorphism $\Phi:\mathbb{P}_1 \to \mathbb{P}_1$ has good reduction at v if there exists a model $\Psi$ for $\Phi$ such that $deg \Psi_v$, the degree of the reduction of $\Psi$ modulo v, equals $deg \Psi$ and $\Psi_v$ is separable. We prove a criterion for good reduction that is the natural generalization of a result due to Zannier in [Z3]. Our result is in connection with other two notions of good reduction, the simple and the critically good reduction. The last part of our article is dedicated to prove a characterization of the maps whose iterates, in a certain sense, preserve the critically good reduction. Jung Kyu Canci2019-03-28T09:51:43Z2019-06-30T17:45:59Zhttps://edoc.unibas.ch/id/eprint/69968This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/699682019-03-28T09:51:43ZOn preperiodic points of rational functions defined over F_p(t)Let $P\in\mathbb{P}_1(\mathbb{Q})$ be a periodic point for a monic polynomial with coefficients in $\mathbb{Z}$. With elementary techniques one sees that the minimal periodicity of P is at most 2. Recently we proved a generalization of this fact to the set of all rational functions defined over $\mathbb{Q}$ with good reduction everywhere (i.e. at any finite place of $\mathbb{Q}$). The set of monic polynomials with coefficients in $\mathbb{Z}$ can be characterized, up to conjugation by elements in $PGL_2(\mathbb{Z})$, as the set of all rational functions defined over $\mathbb{Q}$ with a totally ramified fixed point in $\mathbb{Q}$ and with good reduction everywhere. Let p be a prime number and let $\mathbb{F}_p$ bethe field with p elements. In the present paper we consider rational functions defined over the rational global function field $\mathbb{F}_p$ with good reduction at every finite place. We provesome bounds for the cardinality of orbits in $\mathbb{F}_p\cup\{\infty\}$ for periodic and preperiodic points. Jung Kyu CanciLaura Paladino2017-11-29T10:02:22Z2017-11-29T10:02:22Zhttps://edoc.unibas.ch/id/eprint/51692This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/516922017-11-29T10:02:22ZRational periodic points for quadratic mapsLet K be a number field. Let S be a finite set of places of K containing all the archimedean ones. Let R S be the ring of S-integers of K. In the present paper we consider endomorphisms of ℙ 1 of degree 2, defined over K, with good reduction outside S. We prove that there exist only finitely many such endomorphisms, up to conjugation by PGL 2 (R S ), admitting a periodic point in ℙ 1 (K) of order >3. Also, all but finitely many classes with a periodic point in ℙ 1 (K) of order 3 are parametrized by an irreducible curve. Jung Kyu Canci2017-11-28T08:32:33Z2017-11-28T08:32:33Zhttps://edoc.unibas.ch/id/eprint/51691This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/516912017-11-28T08:32:33ZFinite orbits for rational mapsLet K be a number field and φ ∈ K(z) a rational function. Let S be the set of all archimedean places of K and all non-archimedean places associated to the prime ideals of bad reduction for φ. We prove an upper bound for the length of finite orbits for φ in ℙ1 (K) depending only on the cardinality of S. Jung Kyu Canci2017-11-27T12:59:37Z2017-11-27T12:59:37Zhttps://edoc.unibas.ch/id/eprint/51690This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/516902017-11-27T12:59:37ZCycles for rational maps of good reduction outside a prescribed setLet K be a number field and S a fixed finite set of places of K containing all the archimedean ones. Let RS be the ring of S-integers of K. In the present paper we study the cycles in P1(K) for rational maps of degree ≥2 with good reduction outside S. We say that two ordered n-tuples (P0, P1,… ,Pn−1) and (Q0, Q1,… ,Qn−1) of points of P1(K) are equivalent if there exists an automorphism A ∈ PGL2(RS) such that Pi = A(Qi) for every index i∈{0,1,… ,n−1}. We prove that if we fix two points P0,P1∈P1(K), then the number of inequivalent cycles for rational maps of degree ≥2 with good reduction outside S which admit P0, P1 as consecutive points is finite and depends only on S and K. We also prove that this result is in a sense best possible. Jung Kyu Canci2017-11-01T10:47:18Z2017-11-01T10:47:18Zhttps://edoc.unibas.ch/id/eprint/53276This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/532762017-11-01T10:47:18ZPreperiodic points for rational functions defined over a global field in terms of good reductionLet $phi$ be an endomorphism of the projective line defined over a global field $K.$ We prove a bound for the cardinality of the set of $K$–rational preperiodic points for φ in terms of the number of places of bad reduction. The result is completely new in the function field case and is an improvement of the number field case. Jung Kyu CanciLaura Paladino2017-10-31T10:25:03Z2017-10-31T10:25:03Zhttps://edoc.unibas.ch/id/eprint/53282This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/532822017-10-31T10:25:03ZOn preperiodic points for rational functions defined over $mathbb{F}_p(t)$Let $Pin mathbb(P)_1(mathbb{Q})$ be a periodic point for a monic polynomial with coefficients in $mathbb{Z}$. With elementary techniques one sees that the minimal periodicity of $P$ is at most 2. Recently we proved a generalization of this fact to the set of all rational functions defined over $mathbb{Q}$ with good reduction everywhere (i.e. at any finite place of $mathbb{Q}$). The set of monic polynomials with coefficients in $mathbb{Z}$ can be characterized, up to conjugation by elements in PGL$_2(mathbb{Z}), as the set of all rational functions defined over $mathbb{Q}$ with a totally ramified fixed point in $mathbb{Q}$ and with good reduction everywhere. Let $p$ be a prime number and let $mathbb{F}_p$ be the field with $p$ elements. In the present paper we consider rational functions defined over the rational global function field $mathbb{F}_p(t)$ with good reduction at every finite place. We prove some bounds for the cardinality of orbits in $mathbb{F}_pcup{infty}$ for periodic and preperiodic points. Jung Kyu CanciLaura Paladino2017-10-12T08:35:40Z2017-10-12T08:35:40Zhttps://edoc.unibas.ch/id/eprint/53512This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/535122017-10-12T08:35:40ZQuadratic maps with a periodic critical point of period 2We provide a complete classification of possible graphs of rational preperiodic points of endomorphisms of the projective line of degree 2 defined over the rationals with a rational periodic critical point of period 2, under the assumption that these maps have no periodic points of period at least 7. We explain how this extends results of Poonen on quadratic polynomials. We show that there are exactly 13 possible graphs, and that such maps have at most nine rational preperiodic points. We provide data related to the analogous classification of graphs of endomorphisms of degree 2 with a rational periodic critical point of period 3 or 4. Jung Kyu CanciSolomon Vishkautsan2016-05-03T09:30:51Z2016-06-30T10:59:33Zhttps://edoc.unibas.ch/id/eprint/39401This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/394012016-05-03T09:30:51ZModuli spaces of quadratic rational maps with a marked periodic point of small orderThe surface corresponding to the moduli space of quadratic endomorphisms of P1 with a marked periodic point of order n is studied. It is shown that the surface is rational over Q when n 5 and is of general type for n = 6. An explicit description of the n = 6 surface lets us find several infinite families of quadratic endomorphisms f : P1-> P1 defined over Q with a rational periodic point of order 6. In one of these families, f also has a rational fixed point, for a total of at least 7 periodic and 7 preperiodic points. This is in contrast with the polynomial case, where it is conjectured that no polynomial endomorphism defined over Q admits rational periodic points of order n > 3. Jérémy BlancJung-Kyu CanciNoam D. Elkies2016-04-28T14:43:51Z2016-06-30T11:01:03Zhttps://edoc.unibas.ch/id/eprint/40391This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/403912016-04-28T14:43:51ZPreperiodic points for rational functions defined over a rational function field of characteristic zeroLet k be an algebraically closed field of characteristic zero. Let K be the rational function field K=k(t). Let ϕ be a nonisotrivial rational function in K(z). We prove a bound for the cardinality of the set of K-rational preperiodic points for ϕ in terms of the number of places of bad reduction and the degree d of ϕ. Jung Kyu Canci2015-10-02T10:00:58Z2015-10-02T10:00:58Zhttps://edoc.unibas.ch/id/eprint/38750This item is in the repository with the URL: https://edoc.unibas.ch/id/eprint/387502015-10-02T10:00:58ZOn some notions of good reduction for endomorphisms of the projective lineJung Kyu Canci Peruginelli GiulioDajano Tossici