An iterative construction of irreducible polynomials reducible modulo every prime

Jones, Rafe

doi:10.1016/j.jalgebra.2012.05.020

Cited by 24 publications

(38 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this paper we specialize on irreducibility questions regarding compositional semigroups of polynomials. This kind of question has been addressed in the specific case of semigroups generated by a single quadratic polynomial, see for example in [6,5,11,10], for analogous results related to additive polynomials, see [14,15]. It is worth mentioning that one of these results [11,Lemma 2.5] has been recently used in [16] by the first and the second author of the present paper to prove [12,Conjecture 1.2].…”

Section: Introductionmentioning

confidence: 85%

See 1 more Smart Citation

On Sets of Irreducible Polynomials Closed by Composition

Ferraguti

Micheli

Schnyder

2016

Arithmetic of Finite Fields

View full text Add to dashboard Cite

Abstract. Let S be a set of monic degree 2 polynomials over a finite field and let C be the compositional semigroup generated by S. In this paper we establish a necessary and sufficient condition for C to be consisting entirely of irreducible polynomials. The condition we deduce depends on the finite data encoded in a certain graph uniquely determined by the generating set S. Using this machinery we are able both to show examples of semigroups of irreducible polynomials generated by two degree 2 polynomials and to give some non-existence results for some of these sets in infinitely many prime fields satisfying certain arithmetic conditions.

show abstract

Section: Introductionmentioning

confidence: 85%

“…Since irreducible polynomials play a fundamental role in applications and in the whole theory of finite fields (see for example [1,2,3,4,5,6]), related questions have a long history (see for example [7,8,9,10,11,12,13]). In this paper we specialize on irreducibility questions regarding compositional semigroups of polynomials.…”

Section: Introductionmentioning

confidence: 99%

On Sets of Irreducible Polynomials Closed by Composition

Ferraguti

Micheli

Schnyder

2016

Arithmetic of Finite Fields

View full text Add to dashboard Cite

show abstract

“…It has been of great interest in recent years the study of irreducible polynomials which can be written as composition of degree two polynomials (see for example [1,2,6,8,9,11,13,14,16,17,18]). Such polynomials are also used in other contexts, see for example Rafe Jones' construction of irreducible polynomials reducible modulo every prime [16] or the proof of [3, Conjecture 1.2] in [5]. In this paper we explain how the theory of this class of polynomials completely fits in a general context which allows the use of powerful machinery coming from the theory of finite automata (in characteristic different from 2).…”

Section: Introductionmentioning

confidence: 99%

Irreducible compositions of degree two polynomials over finite fields have regular structure

Ferraguti

Micheli

Schnyder

2018

The Quarterly Journal of Mathematics

View full text Add to dashboard Cite

show abstract

“…As a specific example, Jones[122, Conjecture 6.3] has conjectured that x 2 + 1 is stable over F p if and only if p = 3, i.e., for f (x) = x 2 + 1, the set (19.3) is {3}. Ferraguti[76] has shown that under some Galois-theoretic assumptions, the set of stable primes (19.3) has density zero.…”

mentioning

confidence: 99%

Current trends and open problems in arithmetic dynamics

Benedetto¹,

Ingram²,

Jones³

et al. 2019

Bull. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

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

show abstract

An iterative construction of irreducible polynomials reducible modulo every prime

Abstract: We give a method of constructing polynomials of arbitrarily large degree irreducible over a

Cited by 24 publications

References 21 publications

On Sets of Irreducible Polynomials Closed by Composition

On Sets of Irreducible Polynomials Closed by Composition

Irreducible compositions of degree two polynomials over finite fields have regular structure

Current trends and open problems in arithmetic dynamics

Contact Info

Product

Resources

About