Generalized Büchi’s problem for algebraic functions and meromorphic functions

An, Ta Thi Hoai; Huang, Hsiu-Lien; Wang, Julie Tzu-Yueh

doi:10.1007/s00209-012-0997-9

Cited by 6 publications

(7 citation statements)

References 13 publications

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“…Note that if f (X) ∈ A[X] is a polynomial of degree d ≥ 1 with leading coefficient a, then f (X + 1) − f (X) is a polynomial of degree d − 1 with leading coefficient da. We also observe that applying ∆ (k) to a sequence is the same as applying ∆ (1) a total of k times. These two remarks give the result.…”

Section: Consequences In Logicmentioning

confidence: 81%

“…It is not difficult to produce other exceptional cases for the higher exponents analogue of Büchi's problem in positive characteristic (see [1]). With this consideration in mind, it becomes clear that a satisfactory solution in this context should provide an explicit description of all exceptional cases, and this is the purpose of this section.…”

Section: Generalized Büchi's Problem In Positive Characteristicmentioning

confidence: 99%

“…The only results on Büchi's n-th powers problem for Z (and number fields) beyond the case of squares are the results of [11] which are conditional to a version of the abc conjecture. However, much (unconditional) progress has been achieved in Büchi's n-th powers problem for meromorphic functions and function fields in characteristic zero, and the following result surveys most of what is known (see [1,10,11]):…”

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confidence: 94%

“…More precisely, in [1] it is obtained M K n 5 and their proof also works for p-adic meromorphic functions, while in [11] it is obtained M K n 2 using a different approach, and the result also holds for questions on µ-powerful values of F (i.e. all the zeros of F (b) have multiplicity at least µ) rather than n-th powers.…”

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confidence: 98%

“…However, Büchi's n-th powers problem for one variable function fields in positive characteristic remained mysterious. Even an appropriate formulation of it for n > 2 was unclear due to several unexpected examples of degree n polynomials taking n-th power values in the spirit of [15], but with unknown general pattern (see [1] for some families of such examples in higher exponents). Note, however, that some partial results are known; power values (with higher exponents) of quadratic polynomials have been studied by Garcia-Fritz [4] including the case of positive characteristic, while Hensley's problem in positive characteristic was first considered by Shlapentokh and Vidaux [18], and later solved by Wang [23] when the exponent n is large enough with respect to the genus of the function field (avoiding exceptional examples).…”

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confidence: 99%

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Extensions of Buchi's Higher Powers Problem to Positive Characteristic

Pasten

Wang

2014

International Mathematics Research Notices

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Büchi's n-th power problem on Q asks whether there exist an integer M such that the only monic polynomials F ∈ Q[X] of degree n satisfying that F (1),. .. , F (M) are n-th power rational numbers, are precisely of the form F (X) = (X +c) n for some c ∈ Q. In this paper, we study analogues of this problem for algebraic function fields of positive characteristic. We formulate and prove an analogue (indeed, such a formulation for n > 2 was missing in the literature due to some unexpected phenomena), which we use to derive some definability and undecidability consequences. Moreover, in the case of characteristic zero we extend some known results by improving the bounds for M (from quadratic on n to linear on n). Contents Büchi's undecidability problem. Does there exists an algorithm for the following decision problem? Given diagonal quadratic forms Q j (x 1 ,. .. , x n) = a j1 x 2 1 +. .. + a jn x 2 n (for j = 1,. .. , m) with integer coefficients, and given b = (b j) j ∈ Z m , decide whether or not there is a tuple of integers a ∈ Z n such that for each j = 1,. .. , m we have Q j (a) = b j. More precisely, Büchi formulated an arithmetic problem which, if true, would imply that the previous problem has a negative answer. See [6, 8, 9] for more details. The arithmetic problem formulated by Büchi is the following.

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Section: Consequences In Logicmentioning

confidence: 81%

Section: Generalized Büchi's Problem In Positive Characteristicmentioning

confidence: 99%

mentioning

confidence: 94%

mentioning

confidence: 98%

mentioning

confidence: 99%

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