A Finiteness Result for Post-critically Finite Polynomials

Ingram, Patrick

doi:10.1093/imrn/rnr030

Cited by 24 publications

(54 citation statements)

References 11 publications

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“…and such that for all i, j ∈ X ′ with j > i, we have (9) λ Summing (10) and (11) and applying the product formula, we deduce that where g ∼ f if and only if g is conjugate to f by a Möbius transformation over Q. By the proof of [17,Lemma 6.32], we have…”

Section: Key Inequalitiesmentioning

confidence: 91%

A lower bound on the canonical height for polynomials

Looper¹

2018

Math. Ann.

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We prove a lower bound on the canonical height associated to polynomials over number fields evaluated at points with infinite forward orbit. The lower bound depends only on the degree of the polynomial, the degree of the number field, and the number of places of bad reduction. In the dynamical setting, Silverman has made the following conjecture [15, §4.11].Conjecture 1.1. Let h M d be the height function associated to an embedding of the space of degree d ≥ 2 rational maps M d into projective space, and let K be a number field. There exists a positive constant c depending only on K, d, and the choice of h M d , such that for all rational maps φ ∈ K(z) of degree d,for any non-preperiodic point P ∈ P 1 (K).

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Section: Key Inequalitiesmentioning

confidence: 91%

A lower bound on the canonical height for polynomials

Looper¹

2018

Math. Ann.

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“…The orbits of critical points, and their relation to local behaviour at fixed points, has long been a subject of interest in holomorphic dynamics. The collection of rational functions of a given degree (modulo change of coordinates and ignoring the Lattès examples) with all critical orbits finite turns out to be a set of bounded height, a fact conjectured by Silverman [20] and proven by Benedetto, the author, Jones, and Levy [4] (see also [9,11,12,13,18]). Silverman's conjecture was motivated in part by Thurston's rigidity result for families of post-critically finite rational functions, and Epstein pointed out to the author that there are other, related rigidity results that might suggest similar arithmetic conjectures.…”

Section: Introductionmentioning

confidence: 89%

“…In light of the relation between the moduli height and the critical height established in [11,12], one may view Theorem 1 as showing that on Per n (λ), no single Date: September 10, 2018. 2010 Mathematics Subject Classification.…”

Section: Introductionmentioning

confidence: 99%

“…Before proceeding, we make a few remarks on the proof. The proof in [11] that the collection of PCF polynomials is a set of bounded height is essentially local, constructing an inequality at each place of a number field, and concluding the main result simply by summing those inequalities. The analogous result for rational functions [4] is similarly local, as is proof of the stronger result that the critical height is commensurate to an ample Weil height on the moduli space [12] (although in this case the local inequality is not quite a local version of the global inequality, due to some extra terms which vanish when summing over all places).…”

Section: Introductionmentioning

confidence: 99%

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Critical orbits of polynomials with a periodic point of specified multiplier

Ingram

2018

Math. Z.

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Answering a question posed by Adam Epstein, we show that the collection of conjugacy classes of polynomials admitting a parabolic fixed point and at most one infinite critical orbit is a set of bounded height in the relevant moduli space. We also apply the methods over function fields to draw conclusions about algebraically parametrized families, and prove an analogous result for quadratic rational maps.

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“…For example, when working with polynomials, it is conventional to move the totally ramified fixed point to the point at infinity. Similarly, there are models that are convenient for working with critical points [Ing12] or multipliers [Mil93]. We will use the ideas of [SC03] together with the new bounds from Section 4 to devise an algorithm that finds a model of smallest height in the SL(2, Z)-orbit of a given model.…”

Section: Introductionmentioning

confidence: 99%

Smallest representatives of $\rm {SL}(2,\mathbb {Z})$-orbits of binary forms and endomorphisms of $\mathbb {P}^1$

Hutz

Stoll

2019

Acta Arith.

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We develop an algorithm that determines, for a given squarefree binary form F with real coefficients, a smallest representative of its orbit under SL(2, Z), either with respect to the Euclidean norm or with respect to the maximum norm of the coefficient vector. This is based on earlier work of Cremona and Stoll [SC03]. We then generalize our approach so that it also applies to the problem of finding an integral representative of smallest height in the PGL(2, Q) conjugacy class of an endomorphism of the projective line. Having a small model of such an endomorphism is useful for various computations."Natural" means that this association is compatible with the action of PGL(2) = Aut(P 1 ) on both sides. In this way, we can set up a "reduction theory" for the objects Φ by selecting a suitable "reduced" representative F · γ in the SL(2, Z)-orbit of F and declaring Φ · γ to be the reduced representative in the SL(2, Z)-orbit of Φ. This is the approach taken in [Sto11] in the setting of P n for general n. This works reasonably well when we just want to have a way of selecting a canonical representative of moderate size. If we want to find a representative of smallest height, then more work is required: we have to relate the height of Φ to the size of F(Φ) and determine a bound on the distance we can move from the canonical representative without increasing the height. The application we have in mind is to endomorphisms of P 1 ; this is discussed in some detail in Section 2 below. Main ResultsOur main results are as follows.(1) We prove a result (Theorem 4.7) that bounds the size of a binary form F in terms of its Julia invariant θ(F) (see below) and the hyperbolic distance of its covariant z(F) (see below) to the "center" i of the hyperbolic plane H.(2) Based on this result, we construct an algorithm (Algorithm 5.1) that determines a representative of smallest size in the SL(2, Z)-orbit of a given binary form.(3) We generalize this algorithm so that it can be used with different notions of "size" and for more general objects associated to P 1 . We apply this specifically to automorphisms of P 1 ; see Section 6. (4) In the latter context, we also show how to find minimal representatives up to GL(2, Q)-conjugacy. This involves the determination of all GL(2, Z)-orbits of minimal models; see the algorithms in Section 7. In this context, we prove that an automorphism of even degree has only one orbit of minimal models (Proposition 7.2), but there can be an arbitrary number of orbits for odd degree (Proposition 7.4).

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A Finiteness Result for Post-critically Finite Polynomials

Cited by 24 publications

References 11 publications

A lower bound on the canonical height for polynomials

A lower bound on the canonical height for polynomials

Critical orbits of polynomials with a periodic point of specified multiplier

Smallest representatives of $\rm {SL}(2,\mathbb {Z})$-orbits of binary forms and endomorphisms of $\mathbb {P}^1$

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