Iteration of Quadratic Polynomials Over Finite Fields

Heath‐Brown, D. R.

doi:10.1112/s0025579317000328

Cited by 8 publications

(4 citation statements)

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“…In the context of Conjecture 18.3(a), one might ask instead for a density 1 conclusion for primes whose orbits are smaller than p (1/2+ǫ)N . As explained in [105], it seems unlikely that such a result holds in general, so we pose the problem of characterizing those maps for which there is such a result. There are a handful of published results giving upper bounds for the size of the orbit Of (P mod p), including [105,129,201].…”

Section: Question 182 (Vague Motivating Question) To What Extent Domentioning

confidence: 99%

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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: Question 182 (Vague Motivating Question) To What Extent Domentioning

confidence: 99%

“…There are a handful of published results giving upper bounds for the size of the orbit O f ( P mod p), including [105,129,201]. To illustrate, we quote an easily stated corollary from one of these papers.…”

Section: Question 182 (Vague Motivating Question)mentioning

confidence: 99%

Current trends and open problems in arithmetic dynamics

Benedetto¹,

Ingram²,

Jones³

et al. 2019

Bull. Amer. Math. Soc.

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“…In [HB17], Heath-Brown computed explicit bounds on periodic points of a family of dynamical systems; indeed, for any a, c ∈ Z ≥1 , he showed that…”

Section: Introductionmentioning

confidence: 99%

The Cycle Structure of Unicritical Polynomials

Bridy

Garton

2018

International Mathematics Research Notices

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A polynomial with integer coefficients yields a family of dynamical systems indexed by primes as follows: for any prime p, reduce its coefficients mod p and consider its action on the field F p . The questions of whether and in what sense these families are random have been studied extensively, spurred in part by Pollard's famous "rho" algorithm for integer factorization (the heuristic justification of which is the randomness of one such family). However, the cycle structure of these families cannot be random, since in any such family, the number of cycles of a fixed length in any dynamical system in the family is bounded. In this paper, we show that the cycle statistics of many of these families are as random as possible. As a corollary, we show that most members of these families have many cycles, addressing a conjecture of Mans et. al.

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“…The case of quadratic polynomials has received a special attention, see [10,19,29]. It is easy to see that the functional graph of a quadratic polynomial aX 2 + bX + c is isomorphic to the functional graph of some polynomial X 2 + d (see, for example, the proof of [15,Theorem 2.1]).…”

Section: Introductionmentioning

confidence: 99%

On Functional Graphs of Quadratic Polynomials

Mans

Sha

Shparlinski

et al. 2017

Experimental Mathematics

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We study functional graphs generated by quadratic polynomials over prime fields. We introduce efficient algorithms for methodical computations and provide the values of various direct and cumulative statistical parameters of interest. These include: the number of connected functional graphs, the number of graphs having a maximal cycle, the number of cycles of fixed size, the number of components of fixed size, as well as the shape of trees extracted from functional graphs. We particularly focus on connected functional graphs, that is, the graphs which contain only one component (and thus only one cycle). Based on the results of our computations, we formulate several conjectures highlighting the similarities and differences between these functional graphs and random mappings.2010 Mathematics Subject Classification. 05C20, 05C85, 11T24.

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Iteration of Quadratic Polynomials Over Finite Fields

Cited by 8 publications

References 5 publications

Current trends and open problems in arithmetic dynamics

Current trends and open problems in arithmetic dynamics

The Cycle Structure of Unicritical Polynomials

On Functional Graphs of Quadratic Polynomials

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