Polynomial values modulo primes on average and sharpness of the larger sieve

Shao, Xuancheng

doi:10.2140/ant.2015.9.2325

Cited by 7 publications

(8 citation statements)

References 33 publications

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“…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]. To illustrate, we quote an easily stated corollary from one of these papers.…”

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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“…answering a question of Chowla [3]. Other results in this area have been proven in [10], [21], [8], etc. The connection of these problems to Galois theory is established via the Chebotarev Density Theorem.…”

Section: Introductionmentioning

confidence: 88%

The image size of iterated rational maps over finite fields

Juul¹

2017

Preprint

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Let ϕ : Fq → Fq be a rational map on a fixed finite field. We give explicit asymptotic formulas for the size of image sets ϕ n (Fq) as a function of n. This is done by using properties of the Galois groups of iterated maps, whose connection to the question of the size of image sets is established via Chebotarev's Density Theorem. We then apply these results to provide explicit bounds on the proportion of periodic points in Fq in terms of q for certain rational maps.

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“…Shao [5,Theorem 1.6] handled the case f (X ) = X 2 + 1 by a method which generalizes readily to other quadratics. The condition in his theorem is stronger than ours (that f i (0) = f j (0) for 0 i < j r ) but an examination of the proof shows that he only needs something like our condition.…”

Section: Iteration Of Quadratic Polynomials Over Finite Fields 1043mentioning

confidence: 99%

“…Here we have φ(X a , X b ; d(a, b)) | f d(b,c) (X a )−f d(b,c) (X b ) by (5), since d(a, b) < d(b, c). Hence φ(X a , X c , d(a, c)) is in the ideal generated by φ(X a , X b , d(a, b)) and φ(X b , X c , d(b, c)).…”

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confidence: 95%

Iteration of Quadratic Polynomials Over Finite Fields

Heath‐Brown¹

2017

Mathematika

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Abstract. For a finite field of odd cardinality q, we show that the sequence of iterates of a X 2 +c, starting at 0, always recurs after O(q/log log q) steps. For X 2 +1, the same is true for any starting value. We suggest that the traditional "birthday paradox" model is inappropriate for iterates of X 3 + c, when q is 2 mod 3. §1. Introduction. Let f (X ) ∈ F q [X ] and define the iterates f j (X ) by setting f 0 (X ) = X and f j+1 (X ) = f ( f j (X )). Let m ∈ F q and consider the sequence of values f 0 (m), f 1 (m), f 2 (m), . . . . Since the field F q is finite, the sequence eventually recurs and one enters a closed cycle. We are interested in the questions: how long is it before one enters the cycle? How long is the cycle? In general we can construct a directed graph f = f (F q ), whose vertices are the elements m of F q and with edges (m, f (m)). The trajectory f 0 (m), f 1 (m), f 2 (m), . . . then consists of a pre-cyclic "tail", followed by a cycle.Linear polynomials are easily handled. When f (X ) = X + b, one has f j (X ) = X + bj, so that if b = 0 the cycles are singleton sets and, if b = 0, then f is a union of cycles of length p, the characteristic of the field. For linear polynomials f (X ) = a X + b with a = 0, 1, one finds that

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Polynomial values modulo primes on average and sharpness of the larger sieve

Cited by 7 publications

References 33 publications

Current trends and open problems in arithmetic dynamics

Current trends and open problems in arithmetic dynamics

The image size of iterated rational maps over finite fields

Iteration of Quadratic Polynomials Over Finite Fields

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