Counting rational points on a certain exponential-algebraic surface
Let \({\mathcal X}=\{(x,y,z)\in (0,1)^3:\log x \log y=-\log z\}.\) Let \(F\subset{\mathbb R}\) be a number field of degree \(f\) over \(\mathbb Q\) and let \(\varepsilon>0\). The author proves that there exists a constant \(c({\mathcal X},f,\varepsilon)>0 \) such that \[ \#{\mathcal X}(F,T)\leq c({\mathcal X},f,\varepsilon)(\log T)^{44+\varepsilon}, \] where \({\mathcal X}(F,T)\) denotes the set of \(F\)-rational points of \({\mathcal X}\) with height \(\leq T \).
scientific article; zbMATH DE number 5726202
scientific article