Unlikely Intersection for Two-Parameter Families of Polynomials

Ghioca, Dragos; Hsia, Liang Chung; Tucker, Thomas J.

doi:10.1093/imrn/rnw006

Cited by 3 publications

(4 citation statements)

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“…Since each polynomial g is conjugate with a polynomial in normal form, i.e., there exists a linear polynomial µ such that µ −1 • g • µ is in normal form, one can focus on the dynamics corresponding to polynomials as in Theorem 1.1. In [GHT,Theorem 1.4], the special case m = 2 in Theorem 1.1 was proven, while the case of an arbitrary m was conjectured in [GHT, Question 1.1]. Our Theorem 1.1 answers completely the problem raised in [GHT].…”

Section: Introductionmentioning

confidence: 53%

“…The principle of unlikely intersections for 1-parameter families of rational functions f t predicts that given two starting points c 1 and c 2 which are not persistently preperiodic for the family f , if there exist infinitely many parameters t such that both c 1 and c 2 are preperiodic for f t , then the two starting points are dynamically related; for more details, see [BD11, BD13, GH13, GHT13, GHT15, GHT, GKN16, GKNY, MZ10, MZ12, MZ14]. For higher dimensional families of rational functions, there are very few definitive results, generally limited to 2-parameter families of dynamical systems; see [GHT15,Theorem 1.4] and [GHT,Theorem 1.4]. In this paper we prove the following result regarding unlikely intersections for arithmetic dynamics in higher dimensional parameter spaces.…”

Section: Introductionmentioning

confidence: 99%

“…• for more general families of polynomials and starting points (see [GHT13,BD13,GY18,FG18]), including families of polynomials parameterized by points in a higher dimensional space (see [GHT15,GHT16,GHN18]); • for certain families of rational maps (see [DWY15,GHT15]), and also for arbitrary families of Lattés maps (see [DM20]).…”

Section: Introductionmentioning

confidence: 99%

“…Its proof requires a refined analysis of the valuations for α d −β d , obtained by employing equation (1.5.2) for suitably chosen parameters λ. A similar strategy was previously used for obtaining the precise relation between starting points for other questions regarding the unlikely intersection principle in arithmetic dynamics (see [GHT16,GHN18] in characteristic 0, and also, see [GH13,Ghi24] in characteristic p). However, each time, the technical arguments are different because one has to take into account the special features of the given dynamical system.…”

Section: Introductionmentioning

confidence: 99%

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Simultaneously preperiodic points for families of polynomials in normal form

Ghioca

Hsia

Nguyen

2017

Proc. Amer. Math. Soc.

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Let d > m > 1 be integers, let c 1 , . . . , c m+1 be distinct complex numbers, and let f (z) := z d + t 1 z m−1 + t 2 z m−2 + · · · + t m−1 z + tm be an mparameter family of polynomials. We prove that the set of m-tuples of parameters (t 1 , . . . , tm) ∈ C m with the property that each c i (for i = 1, . . . , m + 1) is preperiodic under the action of the corresponding polynomial f (z) is contained in finitely many hypersurfaces of the parameter space A m .

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Section: Introductionmentioning

confidence: 53%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Simultaneously preperiodic points for families of polynomials in normal form

Ghioca

Hsia

Nguyen

2017

Proc. Amer. Math. Soc.

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Collision of orbits for a one-parameter family of Drinfeld modules

Ghioca

2024

Journal of Number Theory

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Simultaneously preperiodic points for a family of polynomials in positive characteristic

Ghioca

2024

Can. J. Math.-J. Can. Math.

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In the goundbreaking paper [BD11] (which opened a wide avenue of research regarding unlikely intersections in arithmetic dynamics), Baker and DeMarco prove that for the family of polynomials $f_\lambda (x):=x^d+\lambda $ (parameterized by $\lambda \in \mathbb {C}$ ), given two starting points a and b in $\mathbb {C}$ , if there exist infinitely many $\lambda \in \mathbb {C}$ such that both a and b are preperiodic under the action of $f_\lambda $ , then $a^d=b^d$ . In this paper, we study the same question, this time working in a field of characteristic $p>0$ . The answer in positive characteristic is more nuanced, as there are three distinct cases: (i) both starting points a and b live in ${\overline {\mathbb F}_p}$ ; (ii) d is a power of p; and (iii) not both a and b live in ${\overline {\mathbb F}_p}$ , while d is not a power of p. Only in case (iii), one derives the same conclusion as in characteristic $0$ (i.e., that $a^d=b^d$ ). In case (i), one has that for each $\lambda \in {\overline {\mathbb F}_p}$ , both a and b are preperiodic under the action of $f_\lambda $ , while in case (ii), one obtains that also whenever $a-b\in {\overline {\mathbb F}_p}$ , then for each parameter $\lambda $ , we have that a is preperiodic under the action of $f_\lambda $ if and only if b is preperiodic under the action of $f_\lambda $ .

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Unlikely Intersection for Two-Parameter Families of Polynomials

Abstract: Abstract. Let c1, c2, c3 be distinct complex numbers, and let d ≥ 3 be an integer. We show that the set of all pairs (a, b) ∈ C × C such that each ci is preperiodic for the action of the polynomial x d + ax + b is not Zariski dense in the affine plane.

Cited by 3 publications

References 38 publications

Simultaneously preperiodic points for families of polynomials in normal form

Simultaneously preperiodic points for families of polynomials in normal form

Collision of orbits for a one-parameter family of Drinfeld modules

Simultaneously preperiodic points for a family of polynomials in positive characteristic

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