Let \(q\) be a rational number, \(d\) a positive integer, \(r\), \(\delta\), \(E\) positive real numbers, \(f\) an analytic function on an open set containing the disc \[ \{z\in{\mathbb{C}}\; \mid \; |z-q|\leq 4r(1+\delta)\}. \] Suppose \(f(q)\) is a transcendental number with the following transcendence measure: for any \(S\geq e\), any \(T\geq 16\) and any nonzero polynomial \(Q\in{\mathbb{Z}}[X]\) of degree at most \(S\) and coefficients of absolute values at most \(T\), we have \[ \log|Q(f(q))| \geq - E S(S\log S+\log T)(1+\log S). \] Then there exists an effectively computable constant \(C>0\) such that, for all \(H\geq e^e\), there are at most \(C(\log H)^3\log\log H\) complex algebraic numbers \(z\) such that \(|z-q|\leq r\), \({\mathbb{Q}}[(z,f(z)):{\mathbb{Q}}]\leq d\) and \(\max\{ H(z), H(f(z)) \}\leq H\). Here, for an algebraic number \(\alpha\), \(H(\alpha)\) is the absolute multiplicative height of \(\alpha\).
The assumption on the transcendence measure is satisfied when \(f(q)\) is the value of a non constant polynomial with algebraic coefficients at the point \(\pi\). The example of \(f(z)=6(z-2)\zeta(z-1)\pi^{1-z}\) provides an answer to a question raised by J. Pila [Duke Math. J. 63, No. 2, 449–463 (1991; Zbl 0763.11025)]. The authors also prove a variant of the above theorem where no transcendence measure is assumed, but growth conditions are required for \(f\). In particular they prove that if \(r\), \(a\), \(b\), \(s\), \(t\) are positive real numbers with \(b>1\) and \(f\) an entire function satisfying \(|f(z)|<s|z|^{t|z|}\) for all \(|z|\geq r\) and \(f(x)|\leq ab^{-x}\) for all \(x>r\), then for any positive integer \(d\) there exists an effective number \(C>0\) such that, for all \(H>e^e\), there are at most \(C(\log H)^3(\log\log H)^3\) real number \(x>0\) such that \(f(x)\not=0\), \([{\mathbb{Q}}(x,f(x)):{\mathbb{Q}}]\leq d\) and \(\max\{ H(z), H(f(z)) \}\leq H\). As an application, they answer a question of D. Masser on the algebraic values of the Euler Gamma function [J. Number Theory 131, No. 11, 2037–2046 (2011; Zbl 1267.11091)] (see also [E. Besson, Arch. Math. 103, No. 1, 61–73 (2014; Zbl 1303.11079)]).