Definability of Frobenius orbits and a result on rational distance sets

Pasten, Hector

doi:10.1007/s00605-016-0973-2

Cited by 11 publications

(9 citation statements)

References 22 publications

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“…This should be compared to the following result of Pheidas [37] (see [48] for characteristic 2, and see [33] for a generalization to function fields of bounded genus uniformly on the characteristic): Theorem 5.33 (Pheidas). Let k be a finite field of characteristic p > 0.…”

Section: 2mentioning

confidence: 99%

Notes on the DPRM property for listable structures

Pasten¹

2020

Preprint

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The celebrated DPRM theorem by M. Davis, H. Putnam, J. Robinson, and Y. Matiyasevich shows that a set of integers is listable (also called recursively enumerable, or computably enumerable) if and only if it is Diophantine, i.e., positive existentially definable over the language of arithmetic. We present some foundational material to investigate analogues of the DPRM theorem over other structures. This allows us to give rigorous proofs of various folklore results in the literature. In addition, we investigate different conjectures that point in the direction that the analogue of the DPRM theorem for global fields does not hold.

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Section: 2mentioning

confidence: 99%

Notes on the DPRM property for listable structures

Pasten¹

2020

Preprint

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“…Another related question is the Erdős-Ulam conjecture, that there are no everywhere dense pointsets in the plane such that all pairwise distances are rational. This is also open although there are works showing that the existence of such sets would contradict the Bombieri-Lang conjecture [24,21,4] and the abc conjecture as well [18]. In the plane the diameter of large integral pointsets should be large [22,15,3], but not much is known about the structure of such sets.…”

Section: Integral Distancesmentioning

confidence: 99%

Rank of matrices with entries from a multiplicative group

Alon¹,

Solymosi²

2021

Preprint

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“…Shaffaf [15] and Tao [19] independently used the weak Lang conjecture to give a negative answer to this question. Pasten [13] also proved that the abc conjecture implies a negative solution to the Erdős-Ulam problem.…”

Section: Introductionmentioning

confidence: 96%

On the Number of Perfect Triangles with a Fixed Angle

Makhul

2020

Discrete Comput Geom

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Richard Guy asked the following question: can we find a triangle with rational sides, medians and area? Such a triangle is called a perfect triangle and no example has been found to date. It is widely believed that such a triangle does not exist. Here we use the setup of Solymosi and de Zeeuw about rational distance sets contained in an algebraic curve, to show that for any angle 0 < θ < π, the number of perfect triangles with an angle θ is finite.

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Definability of Frobenius orbits and a result on rational distance sets

Cited by 11 publications

References 22 publications

Notes on the DPRM property for listable structures

Notes on the DPRM property for listable structures

Rank of matrices with entries from a multiplicative group

On the Number of Perfect Triangles with a Fixed Angle

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