A survey on Büchi’s problem: new presentations and open problems

Pasten, Hector; Pheidas, Thanases; Vidaux, Xavier

doi:10.1007/s10958-010-0181-x

Cited by 15 publications

(15 citation statements)

References 20 publications

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“…There are many other results highly related to these problems (for example, around Büchi's Problem, or its implications in logic, for subrings of number fields) and that we could have cited here, but we will rather suggest to the interested reader to look for further details in the survey [16], and from a different perspective in the paper [2].…”

Section: Notationmentioning

confidence: 97%

Representation of powers by polynomials and the language of powers

Garcia-Fritz¹

2012

Journal of the London Mathematical Society

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Given a field F and polynomials a and b in F[t], we prove that in general the sets {aλ+b : λ∈F} and {λ2+aλ+b : λ∈F} contain only finitely many powers and find bounds that are uniform from various points of view. We derive from this analysis various first‐order definability and undecidability results. For example, we prove that there is no algorithm to decide whether or not an arbitrary system of linear equations over the integers, together with conditions of the form ‘x is a power’ and of the form ‘x is non‐constant’ on some of the variables, has a solution in F[t].

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Section: Notationmentioning

confidence: 97%

Representation of powers by polynomials and the language of powers

Garcia-Fritz¹

2012

Journal of the London Mathematical Society

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“…We will only mention several related results and refer to [10] for a more complete survey on work in this direction. We first note that Büchi's problem is still open.…”

Section: Mmentioning

confidence: 99%

Hensleyʼs problem for complex and non-Archimedean meromorphic functions

Wang

2011

Journal of Mathematical Analysis and Applications

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Büchi's problem asks if there exists a positive integer M such that all x 1 , . . . , x M ∈ Z satisfying the equations x 2 r − 2x 2 r−1 + x 2 r−2 = 2 for all 3 r M must also satisfy x 2 r = (x + r) 2 for some integer x. Hensley's problem asks if there exists a positive integer M such that, for any integers ν and a, if (ν + r) 2 − a is a square for 1 r M, then a = 0.It is not difficult to see that a positive answer to Hensley's problem implies a positive answer to Büchi's problem. One can ask a more general version of the Hensley's problem by replacing the square by n-th power for any integer n 2 which is called the Hensley's n-th power problem. In this paper we will solve Hensley's n-th power problem for complex meromorphic functions and non-Archimedean meromorphic functions.

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“…They find infinitely many non-trivial such sequences lying on elliptic curves of positive rank. For a recent survey on the problem, we refer the reader to [PPV10].…”

Section: Introductionmentioning

confidence: 99%

On Büchi’s K3 surface

2014

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We study the Büchi K3 surface proving that it belongs to the one dimensional family of Kummer surfaces associated to genus two curves with automorphism group D 4 . We compute its Picard lattice and show that the rational points of the surface are Zariski-dense. Moreover, we provide analogous results for the Kummer surface associated to any genus two curve whose automorphism group contains a non-hyperelliptic involution.

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A survey on Büchi’s problem: new presentations and open problems

Cited by 15 publications

References 20 publications

Representation of powers by polynomials and the language of powers

Representation of powers by polynomials and the language of powers

Hensleyʼs problem for complex and non-Archimedean meromorphic functions

On Büchi’s K3 surface

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