Computing points of small height for cubic polynomials

Benedetto, Robert L.; Dickman, Benjamin; Joseph, Sasha; Krause, Benjamin; Rubin, Daniel; Zhou, Xinwen

doi:10.2140/involve.2009.2.37

Cited by 7 publications

(10 citation statements)

References 19 publications

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“…Thus both f (x) and 12 − f (x) exhibit dynamical compression and have at least 11 rational preperiodic points. These examples share the current record with 8 other cubics, found by computational search in Benedetto et al [3,Table 2], for the cubic polynomial with the most rational preperiodic points. Of the 10 record holding cubics found by Benedetto et al, only the two conjugate to f (x) and 12 − f (x) exhibit dynamical compression.…”

supporting

confidence: 76%

Polynomials with many rational preperiodic points

Doyle¹,

Hyde²

2022

Preprint

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In this paper we study two questions related to exceptional behavior of preperiodic points of polynomials in Q[x]. We show that for all sufficiently large integers d, there exists a polynomialhas at least d + log 16 (d) rational preperiodic points. Furthermore, we show that for infinitely many d ≥ 2, the polynomials f d (x) and f d (x) + 1 have at least d 2 +d log 16 (d) −2d+1 common complex preperiodic points. To prove the existence of such polynomials we introduce the technique of dynamical compression.

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supporting

confidence: 76%

Polynomials with many rational preperiodic points

Doyle¹,

Hyde²

2022

Preprint

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“…(D, N, d) and c 2 (D, N, d) such that for all number fields K/Q with [K : Q] ≤ D, all degree d morphisms f : P N → P N defined over K, and all points P ∈ P N (K) whose orbit O f (P ) is Zariski dense in P N , we haveĥ N, d). See [35,34] for numerical evidence supporting Conjecture 16.3 in the case N = D = 1 and d ∈ {2, 3}, including conjectural values for the constants. We also note that the Zariski density assumption in Conjecture 16.3 is necessary.…”

Section: Conjecture 162 (Dynamical Lehmer Conjecturementioning

confidence: 98%

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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“…We say that a point P ∈ P 1 (K) is preperiodic for f if the orbit of P under f , i.e., the set { f m (P) : m ≥ 0}, is finite. Furthermore, we say that P is periodic for f if it satisfies the stronger condition that f m (P) = P for some m > 0; in this case, the least positive integer m with this property is called the period of P. The set of all points P ∈ P 1 (K) that are preperiodic for f is denoted by PrePer( f , K). Finally, the degree of f (z) is defined to be the number d := max{deg A, deg B}.…”

Section: Introductionmentioning

confidence: 99%

“…However, a proof of this conjecture seems distant at present. One direction in which to continue the kind of work carried out by Poonen in studying the Uniform Boundedness Conjecture (UBC) is to consider preperiodic points for higher degree polynomials over Q, such as was done by Benedetto et al [1] in the case of cubics. In this paper we take a different approach and consider maps defined over number fields of degree n > 1.…”

Section: Introductionmentioning

confidence: 99%

Preperiodic points for quadratic polynomials with small cycles over quadratic fields

Doyle

2017

Math. Z.

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Given a number field K and a polynomial ϕ(z) ∈ K[z] of degree at least 2, one can construct a finite directed graph G(ϕ, K) whose vertices are the K-rational preperiodic points for f , with an edge α → β if and only if ϕ(α) = β. Restricting to quadratic polynomials, the dynamical uniform boundedness conjecture of Morton and Silverman suggests that for a given number field K, there should only be finitely many isomorphism classes of directed graphs that arise in this way. Poonen has given a conjecturally complete classification of all such directed graphs over Q, while recent and ongoing work of the author, Faber, Krumm, and Wetherell has provided a detailed study of this question for all quadratic extensions of Q. In this article, we give a conjecturally complete classification like Poonen's, but over the cyclotomic quadratic fields Q( √ −1) and Q( √ −3). The main tools we use are dynamical modular curves and results concerning quadratic points on curves.Proof of Theorem 4.6. For (A), we note that 3 is a prime of good reduction for J 1 (4), and thatMoreover, since 5 is a prime of good reduction and #J 1 (4)(F 5 2 ) = 2 7 · 5, J 1 (4)(K) cannot have 3-torsion. Hence J 1 (4)(K) tors → (Z/2Z) 3 ⊕ Z/10Z.Since J 1 (4)(Q) tors ∼ = Z/2Z ⊕ Z/10Z, the only way for J 1 (4)(K) tors to be strictly larger than J 1 (4)(Q) tors is for J 1 (4) to gain a 2-torsion point upon base change from Q to K. By Lemma 3.1,

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Computing points of small height for cubic polynomials

Cited by 7 publications

References 19 publications

Polynomials with many rational preperiodic points

Polynomials with many rational preperiodic points

Current trends and open problems in arithmetic dynamics

Preperiodic points for quadratic polynomials with small cycles over quadratic fields

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