Generalized Büchi's problem for algebraic functions and meromorphic functions
The paper under review analyzes Büchi\'{}s problem for \(n\)-th powers over function fields of characteristic 0 and fields of meromorphic functions. Büchi's problem was originally stated for \(n=2\) and for rings of integers. More specifically the question concerned sequences of integers \(\{x_k\}\) satisfying \(x_{k+2}^2-2x_{k+1}^2 + x_k^2=2\) and existence of a constant \(M\) such that any sequence as described above of length greater or equal to \(M\) must be of the form \(x_n=x +n\) for some \(x \in \mathbb Z\). This problem is still open. The positive answer to this question would lead to a stronger version of the negative answer to Hilbert's Tenth Problem (the question of decidability of Diophantine equations). The original problem has been generalized in several ways: the squares have been replaced by an arbitrary power \(n\) with the condition that the \(n\)-th difference is equal to \(n!\), and the ring of integers has been replaced by other commutative rings. Function field generalizations of this problem have been obtained by \textit{H. Pasten} [Proc. Am. Math. Soc. 138, No. 5, 1549--1557 (2010; Zbl 1222.11146); J. Number Theory 133, No. 9, 2964--2998 (2013; Zbl 1364.11090)], [\textit{T. Pheidas} and \textit{X. Vidaux}, J. Lond. Math. Soc., II. Ser. 74, No. 3, 545--565 (2006; Zbl 1109.03032); corrigendum 82, No. 1, 273--278 (2010); Pac. J. Math. 238, No. 2, 349--366 (2008; Zbl 1151.03004)], [\textit{A. Shlapentokh} and \textit{X. Vidaux}, J. Algebra 330, No. 1, 482--506 (2011; Zbl 1250.11103)], [\textit{P. Vojta}, Contemp. Math. 270, 261--274 (2000; Zbl 0995.11070)] and [\textit{J. T. Y. Wang}, Int. J. Number Theory 8, No. 2, 507--524 (2012; Zbl 1261.11035)], [\textit{T. T. H. An} and \textit{J. T. Y. Wang}, J. Math. Anal. Appl. 381, No. 2, 661--677 (2011; Zbl 1246.30076)], and others. In this paper, the generalized problem is considered and solved over rings and fields of meromorphic and algebraic functions of characteristic 0. The lower bound on the constant \(M\) is given in terms of \(n\) and the genus of the field.
scientific article; zbMATH DE number 6134696
scientific article