Arithmetic properties of rational fractions with algebraic coefficients (Q1591619)

The aim of this paper is to give sufficient conditions on a set \(E \subset\mathbb{Z}^2\) for the following condition to hold: \(f(E) \subset \mathbb{Z}\) implies that the rational function \(f\in\mathbb{Q}(X,Y)\) is polynomial. The main result of this paper states the following sufficient condition on the set \(E\): there exists \(\varepsilon>0\) such that \[ \lim\sup {\text{card} E \cap\bigl\{ (m,n) \in\mathbb{Z}^2,- k\leq m,n\leq k\bigr\}\over k^{3/2+ \varepsilon}}>0. \] This condition ameliorates a result by [\textit{M. Yasumoto}, Manuscr. Math. 85, 1-10 (1994; Zbl 0822.12004)] based on nonstandard models of arithmetic, whereas the proofs in the paper under review use elementary tools. More precisely, the proofs are based on an estimate of the number of integer values taken by a rational function \(f\). An explicit bound \(c(f)\) is first given such that if a rational fraction \(f\) with one indeterminate takes more than \(c(f)\) integer values, \(f\) is a polynomial. The study is then reduced by the specialisation \(Y= aX+b\) to the case of rational fractions with one indeterminate. Note that these results are also extended to rational fractions with algebraic coefficients. The paper ends with a discussion on the best exponent of \(k\) one can get in the above mentioned condition on \(E\). In particular, it is conjectured that the condition holds with \(1+\varepsilon\) whereas a counterexample for the growth order \(k\log k\) is given.