A lower bound on the canonical height for polynomials

Looper, Nicole

doi:10.1007/s00208-018-1735-3

Cited by 5 publications

(5 citation statements)

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“…For example, the map f = [X 2 , Y 2 , γZ 2 ] has height roughly h(γ), while the f -orbit of the point P = [α, 1, 0] has canonical heightĥ f (P ) = h(α), so we can make h M (f ) arbitrarily large whileĥ f (P ) remains bounded. Ingram [116] and Looper [155] have proven weak versions of Conjecture 16.3 for polynomial maps f of P 1 , where the constants depend also on the number of primes of bad reduction of f .…”

Section: Conjecture 162 (Dynamical Lehmer Conjecturementioning

confidence: 99%

Current trends and open problems in arithmetic dynamics

Benedetto¹,

Ingram²,

Jones³

et al. 2019

Bull. Amer. Math. Soc.

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Arithmetic dynamics is the study of number theoretic properties of dynamical systems. A relatively new field, it draws inspiration partly from dynamical analogues of theorems and conjectures in classical arithmetic geometry, and partly from p-adic analogues of theorems and conjectures in classical complex dynamics. In this article we survey some of the motivating problems and some of the recent progress in the field of arithmetic dynamics. 1 2 BENEDETTO, ET AL. 22. Local-Global Questions in Dynamics 61 References 63 TRENDS AND PROBLEMS IN ARITHMETIC DYNAMICS 3background on dynamical systems, number theory, and algebraic geometry that we will need. As noted at the end of Section 3, we have attempted to make this article accessible by generally restricting attention to maps of projective space, thereby obviating the need for more advanced material from algebraic geometry. Our survey of arithmetic dynamics then commences in Section 4 and concludes in Section 22. These nineteen sections are mostly independent of one another, albeit liberally sprinkled with instructive cross-references.A Brief Timeline of the Early Days. The study of Galois groups of iterated polynomials was initiated by Odoni in a series of papers [177,178,179] during the mid-1980s. Beyond this, it appears that the first paper to describe dynamical analogues for a wide range of classical problems in arithmetic geometry did so not for polynomial maps or projective space, but rather for K3 surfaces admitting an automorphism of infinite order. The 1991 paper of Silverman [204] discusses dynamical analogues of various problems, including: (1) Uniform boundedness of periodic points;(2) Periodic points on subvarieties;(3) Lower bounds for canonical heights; (4) Open image of Galois groups; (5) Integral points in orbits. The rest of the 1990s saw an explosion of papers in which numerous researchers began to study a wide variety of problems in arithmetic dynamics. The serious study of p-adic dynamics starts with a 1983 paper of Herman and Yoccoz [107], two of the world's leading complex dynamicists, in which they investigate a p-adic analogue of a classical problem in complex dynamics. Later in the 1980s and 90s, the physics literature includes several papers [6,7,21,230] that discuss p-adic dynamics for polynomial maps, but the deeper study of p-adic dynamics really took off with the Ph.D. theses of Benedetto [24, 25] in 1998 and Rivera-Letelier [190, 191] in 2000. The dynamics of iterated p-adic power series was investigated by Lubin [156] in a 1994 paper. We also mention that there is an extensive literature on ergodic theory over p-adic fields which, although not within the purview of this survey, dates back to at least 1975 [180]. Abstract Dynamical SystemsA discrete dynamical system consists of a set S and a self-map

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Section: Conjecture 162 (Dynamical Lehmer Conjecturementioning

confidence: 99%

Current trends and open problems in arithmetic dynamics

Benedetto¹,

Ingram²,

Jones³

et al. 2019

Bull. Amer. Math. Soc.

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“…Besides the input of height machinery and bounds on heights of preperiodic points of polynomial functions (e.g. from [27,37]), one of the other main ideas is to show that the prime factors of differences of preperiodic points z i − z j are typically in the places of bad reduction when f has many preperiodic points. The abcd conjecture (Conjecture 2.9) can then be applied to combinatorial "polygons" constructed from preperiodic points giving points on the projective hyperplanes H. Following Looper [38] with some case-by-case calculations, Panraksa [54] showed that the usual abc conjecture implies that there are no rational non-fixed periodic points for largedegree unicritical polynomials.…”

Section: Conjecture 35 (Dynamical Uniform Boundedness Conjecturementioning

confidence: 99%

The abcd conjecture, uniform boundedness, and dynamical systems

Zhang

2024

Publications Mathématiques De Besançon. Algèbre Et Théorie Des Nombres

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We survey Vojta's higher-dimensional generalizations of the abc conjecture and Szpiro's conjecture as well as recent developments that apply them to various problems in arithmetic dynamics. In particular, the "abcd conjecture" implies a dynamical analogue of a conjecture on the uniform boundedness of torsion points and a dynamical analogue of Lang's conjecture on lower bounds for canonical heights.Résumé. -Nous décrivons des généralisations en dimension supérieure dues à Vojta de la conjecture abc et de la conjecture de Szpiro, ainsi que des avancées récentes qui les utilisent dans des problèmes variés de dynamique arithmétique. En particulier, la « conjecture abcd » implique un analogue dynamique de la conjecture de torsion et un analogue dynamique de la conjecture de Lang sur les minorations de hauteurs canoniques.

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“…Besides the input of height machinery and bounds on heights of preperiodic points of polynomial functions (e.g. from [Ing09,Loo19]), one of the other main ideas is to show that the prime factors of differences of preperiodic points z i − z j are typically in the places of bad reduction when f has many preperiodic points. The abcd conjecture (Conjecture 2.9) can then be applied to combinatorial "polygons" constructed from preperiodic points giving points on the projective hyperplanes H.…”

Section: Uniform Boundedness Of Preperiodic Pointsmentioning

confidence: 99%