The analogue of Buchi's Problem for function fields

作者：Shlapentokh Alexandra; Vidaux Xavier^*

来源：Journal of Algebra, 2011, 330(1): 482-506.

DOI：10.1016/j.jalgebra.2011.01.008

• Buchi's Problem

'Buchi'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(n)(2) - 2x(n-1)(2) + x(n-2)(2) = 2 for all n), satisfies x(n)(2) = (x + n)(2) for some integer x. Hensley's Problem for r-th powers (where r is an integer >= 2) is a generalization of Biichi's Problem asking for an integer M such that, given integers v and a. the quantity (nu + 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 Buchi's Problem has a positive answer if p >= 312g + 169 (improving results by Pheidas and the second author).

• 出版日期2011-3-15