Hilbert's irreducibility theorem and \(G\)-functions
Let \(k\) be a number field and \(f\in k[X,Y]\) be an absolutely irreducible polynomial with \(n:=\deg_{Y}(P)\geq1\). Let \(Y\) be a rational function on the curve defined by \(f\). Hilbert's irreducibility theorem is restated as follows. For infinitely many \(\xi\in k\) the values \(1\), \(Y(\xi)\), \(\ldots\), \(Y^{n-1}(\xi)\) are \(k\)-linearly independent. The main purpose of this paper is to show that such a linear independent result holds in general for vectors \(\mathbf Y:=(1,Y(\xi),\ldots,Y^{n-1}(\xi))\) of \(G\)-functions satisfying similar conditions. Let \(K\) be a number field containing \(k\) and the coefficients of \(G\)-functions \(Y_1(X)\), \(\ldots\), \(Y_n(X)\). Let \(R_v\) denote the (non-zero) \(v\)-adic radius of convergence of \(\mathbf Y=(Y_1(X),\ldots,Y_n(X))\) for each valuation \(v\) of \(K\). The authors prove the following. If \(Y_1(X)\), \(\ldots\), \(Y_n(X)\) are linearly independent over \(\overline{\mathbb Q}(X)\) then, for every positive number \(R<R_v\), there exist infinitely many \(\xi\in\mathbb Q\cap\{\xi\in K_v| 0\leq| \xi| _v<R_v\}\) such that the elements \(Y_{i,v}(\xi)\) of \(K_v\), \(i=1,\ldots,n\) are linearly independent over \(K\). The proof provides a more precise result.
scientific article; zbMATH DE number 1096013