Solving genus zero Diophantine equations over number fields

Alvanos, Paraskevas; Poulakis, Dimitrios

doi:10.1016/j.jsc.2010.09.002

Cited by 3 publications

(14 citation statements)

References 8 publications

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“…and we are going to develop an algorithm heavily based on the Alvanos and Poulakis' algorithm INTEGRAL-POINTS2A from the paper [3]. In fact, the algorithm presented here is just the application of INTEGRAL-POINTS2A to the curve G.…”

Section: The General Algorithm Over Number Fieldsmentioning

confidence: 99%

“…For this task we will use mainly and heavily the research of Poulakis on solving genus zero Diophantine equations, developed in successive papers with Alvanos, Bilu and Voskos [2,3,31]. In section 3, we present an algorithm based on Alvanos and Poulakis' work [3] that allows us to compute G (a,b,c,d) (O K ). We show how the algorithm described in this section works on some examples not covered by the theoretical results from section 4.…”

Section: Introductionmentioning

confidence: 99%

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Markoff–Rosenberger triples in geometric progression

González–Jiménez

2013

Acta Math Hung

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Section: The General Algorithm Over Number Fieldsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Markoff–Rosenberger triples in geometric progression

González–Jiménez

2013

Acta Math Hung

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“…This situation makes particularly simple the use of the INTEGRAL-POINTS algorithm by Alvanos and Poulakis [1]. What makes this algorithm more remarkable is the fact that it works for arbitrary number fields computing affine points whose coordinates lie in the corresponding ring of integers.…”

Section: Variations On the Markoff Equationmentioning

confidence: 99%

“…But first, as the INTEGRAL-POINTS algorithm will be heavily used in what follows, we will recall its steps, for the convenience of the reader. The proofs concerning correctness and termination, as well as many other interesting features can be found in the original reference [1].…”

Section: Variations On the Markoff Equationmentioning

confidence: 99%

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Markoff–Rosenberger triples in arithmetic progression

González–Jiménez

Tornero

2013

Journal of Symbolic Computation

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We study the solutions of the Rosenberg-Markoff equation ax 2 + by 2 + cz 2 = dxyz (a generalization of the well-known Markoff equation). We specifically focus on looking for solutions in arithmetic progression that lie in the ring of integers of a number field. With the help of previous work by Alvanos and Poulakis, we give a complete decision algorithm, which allows us to prove finiteness results concerning these particular solutions. Finally, some extensive computations are presented regarding two particular cases: the generalized Markoff equation x 2 + y 2 + z 2 = dxyz over quadratic fields and the classic Markoff equation x 2 + y 2 + z 2 = 3xyz over an arbitrary number field.

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Abelian dynamical Galois groups for unicritical polynomials

Ferraguti¹,

Pagano²

2023

Preprint

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Andrews and Petsche proposed in 2020 a conjectural characterization of all pairs (f, α), where f is a polynomial over a number field K and α ∈ K, such that the dynamical Galois group of the pair (f, α) is abelian. In this paper we focus on the case of unicritical polynomials f , and more general dynamical systems attached to sequences of unicritical polynomials.After having reduced the conjecture to the post-critical finite case, we establish it for all polynomials with periodic critical orbit, over any number field. We next establish the conjecture in full for all unicritical polynomials over any quadratic field. Finally we show that for any given degree d there exists a finite, explicit set of unicritical polynomials that depends only on d, such that if f is a unicritical polynomial over a number field K that lies outside such exceptional set, then there are at most finitely many basepoints α such that the dynamical Galois group of (f, α) is abelian.To obtain these results, we exploit in multiple ways the group theory of the generic dynamical Galois group to force diophantine relations in dynamical quantities attached to f . These relations force in all cases, outside of the ones conjectured by Andrews-Petsche, a contradiction either with lower bounds on the heights in abelian extensions, in the style of Amoroso-Zannier, or with the computation of rational points on explicit curves, carried out with techniques from Balakrishnan-Tuitman and Siksek.

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Solving genus zero Diophantine equations over number fields

Cited by 3 publications

References 8 publications

Markoff–Rosenberger triples in geometric progression

Markoff–Rosenberger triples in geometric progression

Markoff–Rosenberger triples in arithmetic progression

Abelian dynamical Galois groups for unicritical polynomials

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