An extension of Büchi’s problem for polynomial rings in zero characteristic

Pasten, Hector

doi:10.1090/s0002-9939-09-10259-9

Cited by 9 publications

(12 citation statements)

References 14 publications

(15 reference statements)

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“…(13) As long as the extension of K over F (t) is separable, we can define a global derivation with respect to t. Over F (t), we use the usual definition of the derivative, and we use the implicit differentiation to extend derivation to the extension (see for example Mason [11,p. 9 and p. 94]).…”

Section: Technical Preliminariesmentioning

confidence: 99%

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The analogue of Büchi's Problem for function fields

Shlapentokh

Vidaux

2011

Journal of Algebra

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Büchi's n Squares Problem asks for an integer M such that any sequence (x 0 , . . . , x M−1 ), whose second difference of squares is the constant sequence (2) (i.e. x 2 n − 2x 2 n−1 + x 2 n−2 = 2 for all n),Büchi's Problem asking for an integer M such that, given integers ν and a, the quantity (ν + n) r − a cannot be an r-th power for M or more values of the integer n, unless a = 0. The analogues of these problems for rings of functions consider only sequences with at least one non-constant term. Let K be a function field of a curve of genus g. We prove that Hensley's Problem for r-th powers has a positive answer for any r if K has characteristic zero, improving results by Pasten and Vojta.In positive characteristic p we obtain a weaker result, but which is enough to prove that Büchi's Problem has a positive answer if p 312g + 169 (improving results by Pheidas and the second author).

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Section: Technical Preliminariesmentioning

confidence: 99%

“…For a general discussion on the equivalence between B 2 (R) and HP 2 (R) (for some rings R the two problems may not be equivalent) see the survey [15], or [13].…”

Section: Introductionmentioning

confidence: 99%

The analogue of Büchi's Problem for function fields

Shlapentokh

Vidaux

2011

Journal of Algebra

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“…In the same paper, he also proved the split function field case of characteristic zero and the case of holomorphic curves. Motivated by [11] for solving Büchi's problem for higher powers, Pasten proposed Hensley's n-th power problem in [8], and solved for polynomial rings of zero characteristic. The case of function fields of zero characteristic was worked out by Shlapentokh and Vidaux in [16] where they also solved Büchi's problem (for squares) in large enough characteristic.…”

Section: Mmentioning

confidence: 99%

“…Recently, the second named author [19] solved the case of function fields in any characteristic. The general method in [8,16,19] and our paper was first introduced by Pheidas and Vidaux in [12] and [13] where they solved Büchi's problem for rational functions in characteristic zero or large enough characteristic. In this paper, we will input techniques from Nevanlinna theory to study the problem for complex and non-Archimedean meromorphic functions and also to treat the case of positive characteristic in the non-Archimedean situation.…”

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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“…Remark. Very recently, H. Pasten [2008] proved a strong version of Büchi's problem for squares over polynomial rings. His result gives new evidence that the analogous problem for any (fixed) power could have a positive answer.…”

Section: Corollary 14 (Undecidability Of Simultaneous Representationmentioning

confidence: 99%

The analogue of Büchi’s problem for cubes in rings of polynomials

Pheidas¹,

Vidaux²

2008

Pacific J. Math.

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Let F be a field of characteristic zero. We give the following answer to a generalization of a problem of Büchi over F [t]: A sequence of 92 or more cubes in F [t], not all constant, with constant third difference equal to 6, consists of cubes of successive elements x, x+1, . . . , for some x ∈ F[t]. We use this, in conjunction to the negative answer to Hilbert's tenth problem for F [t], to show that the solvability of systems of degree-one equations, where some of the variables are assumed to be cubes and (or) nonconstant, is an unsolvable problem over F [t].

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An extension of Büchi’s problem for polynomial rings in zero characteristic

Cited by 9 publications

References 14 publications

The analogue of Büchi's Problem for function fields

The analogue of Büchi's Problem for function fields

Hensleyʼs problem for complex and non-Archimedean meromorphic functions

The analogue of Büchi’s problem for cubes in rings of polynomials

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