Galois Groups of Periodic Points

Morton, Patrick

doi:10.1006/jabr.1997.7304

Cited by 13 publications

(13 citation statements)

References 8 publications

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“…We postpone the discussion of wreath products (which appear in Morton's Theorem as C d ≀S r k (d) ) and their natural action (on the sets B (n, r k (n))) until Section 3, which immediately follows the statement of the theorem; there, we will introduce wreath products and their actions, then discuss them in detail. The following theorem combines results from [Mor98].…”

Section: Dynatomic Polynomialsmentioning

confidence: 76%

“…As we intend to study the cycle structure of dynamical systems induced by polynomials, we will make use of the theory of dynatomic polynomials (and their Galois groups). See [MP94], [Mor96] (and the correction in [Mor11]), [Mor98], and [Sil07, Chapter 4.1] for background in this area. We sketch an introduction, focusing on the aspects of the theory we will use in our results.…”

Section: Dynatomic Polynomialsmentioning

confidence: 99%

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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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Section: Dynatomic Polynomialsmentioning

confidence: 76%

Section: Dynatomic Polynomialsmentioning

confidence: 99%

The Cycle Structure of Unicritical Polynomials

Bridy

Garton

2018

International Mathematics Research Notices

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“…As we intend to distinguish dynamical systems by analyzing their periodic points, we will make use of the theory of dynatomic polynomials (and their Galois groups). See [MP94], [Mor96] (and the correction in [Mor11]), [Mor98], and [Sil07, Chapter 4.1] for background in this area. We sketch an introduction, focusing on the aspects of the theory we will use in our results.…”

Section: Galois Groups Of Dynatomic Polynomialsmentioning

confidence: 99%

“…Proof. Following the proof of Theorem 10 in [Mor98], for any n ∈ Z >0 , there exists a polynomial δ n (x) ∈ Z[x] such that the finite primes in Q(c) that ramify in Σ f,n have the form…”

Section: Galois Groups Of Dynatomic Polynomialsmentioning

confidence: 99%

Dynamically distinguishing polynomials

Bridy

Garton

2017

Res Math Sci

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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 . We say a subset of Z[x] is dynamically distinguishable mod p if the associated mod p dynamical systems are pairwise non-isomorphic. For any k, M ∈ Z >1 , we prove that there are infinitely many sets of integers M of size M such that x k + m | m ∈ M is dynamically distinguishable mod p for most p (in the sense of natural density). Our proof uses the Galois theory of dynatomic polynomials largely developed by Morton, who proved that the Galois groups of these polynomials are often isomorphic to a particular family of wreath products. In the course of proving our result, we generalize Morton's work and compute statistics of these wreath products.

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“…If X ⊂ P n K is an irreducible subvariety such that f r (X) = X for some r > 0, then X is a point or X = P n K . This is a rather straightforward consequence of the transitivity of the monodromy action on the set of periodic points of a fixed period of a generic endomorphism, which we prove (Proposition 3.3) using a result of Bousch [3], Lau-Schleicher [11] and Morton [12] for polynomials in one variable of the form z → z d + c.…”

Section: Introductionmentioning

confidence: 99%

The algebraic dynamics of generic endomorphisms of ℙⁿ

Fakhruddin

2014

Algebra Number Theory

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Abstract. We investigate some general questions in algebraic dynamics in the case of generic endomorphisms of projective spaces over a field of characteristic zero. The main results that we prove are that a generic endomorphism has no non-trivial preperiodic subvarieties, any infinite set of preperiodic points is Zariski dense and any infinite subset of a single orbit is also Zariski dense, thereby verifying the dynamical "Manin-Mumford" conjecture of Zhang and the dynamical "Mordell-Lang" conjecture of Denis and Ghioca-Tucker in this case.

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Galois Groups of Periodic Points

Cited by 13 publications

References 8 publications

The Cycle Structure of Unicritical Polynomials

The Cycle Structure of Unicritical Polynomials

Dynamically distinguishing polynomials

The algebraic dynamics of generic endomorphisms of ℙⁿ

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Galois Groups of Periodic Points

Cited by 13 publications

References 8 publications

The Cycle Structure of Unicritical Polynomials

The Cycle Structure of Unicritical Polynomials

Dynamically distinguishing polynomials

The algebraic dynamics of generic endomorphisms of ℙn

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Product

Resources

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The algebraic dynamics of generic endomorphisms of ℙⁿ