The Galois Theory of Periodic Points of Polynomial Maps

Morton, Patrick; Patel, Pratiksha

doi:10.1112/plms/s3-68.2.225

Cited by 40 publications

(41 citation statements)

References 2 publications

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“…This last phenomenon was already observed for polynomial 0's in [12]. In particular, it is precisely the rationally indifferent periodic point which may appear in some Z*(φ) with n not equal to their period; and if K has characteristic zero, then such a point appears in at most two Z* (φ)'s.…”

Section: ) This Follows Immediately From (A) and The Definition Of Z*supporting

confidence: 53%

“…In this case, (11) gives a$(φ, n) = α ρ (ψ, 1), so we get ifr = l,from (12). In this case, (11) gives a$(φ, n) = α ρ (ψ, 1), so we get ifr = l,from (12).…”

Section: ) This Follows Immediately From (A) and The Definition Of Z*mentioning

confidence: 90%

“…As suggested in [12] and by analogy with the case of cyclotomic polynomials, we define the cycle of essential n-periodic points of φ by the formula d\n Here μ is the M bius function. As suggested in [12] and by analogy with the case of cyclotomic polynomials, we define the cycle of essential n-periodic points of φ by the formula d\n Here μ is the M bius function.…”

Section: For Certain Integers a P (N) Our Remarks Above Make It Cleamentioning

confidence: 99%

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Periodic points, multiplicities, and dynamical units.

Silverman¹,

Morton²

1995

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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at ProvidenceLet φ (z) e C [z] be a polynomial of degree at least 2. The fixed points of the iterates of φ have been widely studied since the time of Julia and Fatou in order to analyze the dynamical System associated to φ. (See, for example, [2].) If φ (ζ) has coefficients in a number field K, these periodic points will generate interesting algebraic extensions of K. More precisely, if α e K is a periodic point for φ, then the map σ : α ι-> φ (α) will frequently induce a field automorphism of K(v) over K. In other words, there is often an element σ e Aut(K(aC)/K) whose action on the generator α is given by the polynomial φ. For a discussion of when φ induces such a σ, see for example [10], [11], [12].The fields generated by periodic points of polynomials, or more generally rational functions φ (z) e K(z), are thus of interest because one is able to express part of the action of the Galois group in terms of explicit polynomial actions. For example, if φ (ζ) = z d , then the periodic points are roots of unity, and one knows that the action of the Galois group on a root of unity is given by a polynomial map of the form z -> z*. Similarly, let [«] : E -> E be the multiplication-by-w map on a CM elliptic curve. Then the periodic points are points of finite order on E, and the action of the Galois group on these points can be expressed in terms of certain rational functions coming from the group law on E.In these last two examples, an important discovery was that in the fields generated by periodic points, one can use the periodic points to produce units. The units arising from roots of unity are called cyclotomic units and are quite classical. The units coming from points of finite order on CM elliptic curves are known s elliptic units and were originally constructed by Robert [18]. (And recently, units have similarly been constructed using division points on Jacobians of genus two curves [3], [7].) It would be hard to overstate the importance of cyclotomic and elliptic units in modern algebraic number theory and arithmetic geometry.

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Section: ) This Follows Immediately From (A) and The Definition Of Z*supporting

confidence: 53%

“…In this case, (11) gives a$(φ, n) = α ρ (ψ, 1), so we get ifr = l,from (12). In this case, (11) gives a$(φ, n) = α ρ (ψ, 1), so we get ifr = l,from (12).…”

Section: ) This Follows Immediately From (A) and The Definition Of Z*mentioning

confidence: 90%

Section: For Certain Integers a P (N) Our Remarks Above Make It Cleamentioning

confidence: 99%

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Periodic points, multiplicities, and dynamical units.

Silverman¹,

Morton²

1995

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…Let F = K (α) be a cubic Galois extension of K . A non-trivial polynomial f (X) ∈ K [X] is called an automorphism polynomial of α and F , respectively, if f (α) = σ (α) for some non-trivial Galois automorphism σ ∈ Gal(F /K ) (see [9,10]). Since F /K is cubic and Galois, each generator of F has exactly two at most quadratic automorphism polynomials, one for σ and one for σ 2 .…”

Section: Arbitrary Ground Fieldmentioning

confidence: 99%

On cubic Galois field extensions

Häberle

2010

Journal of Number Theory

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We study Morton's characterization of cubic Galois extensions F /K by a generic polynomial depending on a single parameter s ∈ K . We show how such an s can be calculated with the coefficients of an arbitrary cubic polynomial over K the roots of which generate F . For K = Q we classify the parameters s and cubic Galois polynomials over Z, respectively, according to the discriminant of the extension field, and we present a simple criterion to decide if two fields given by two s-parameters or defining polynomials are equal or not.

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“…8, Sec. 3.4͒.The periodic points of essential period T of a polynomial f (x) are the roots of the polynomial[21][22][23][24] …”

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confidence: 99%

Complexity of regular invertible p-adic motions

Pettigrew

Roberts

Vivaldi

2001

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We consider issues of computational complexity that arise in the study of quasi-periodic motions (Siegel discs) over the p-adic integers, where p is a prime number. These systems generate regular invertible dynamics over the integers modulo p(k), for all k, and the main questions concern the computation of periods and orbit structure. For a specific family of polynomial maps, we identify conditions under which the cycle structure is determined solely by the number of Siegel discs and two integer parameters for each disc. We conjecture the minimal parametrization needed to achieve-for every odd prime p-a two-disc tessellation with maximal cycle length. We discuss the relevance of Cebotarev's density theorem to the probabilistic description of these dynamical systems. (c) 2001 American Institute of Physics.

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The Galois Theory of Periodic Points of Polynomial Maps

Cited by 40 publications

References 2 publications

Periodic points, multiplicities, and dynamical units.

Periodic points, multiplicities, and dynamical units.

On cubic Galois field extensions

Complexity of regular invertible p-adic motions

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