Transcendental divisors and their critical functions
The main purpose of the paper under review is to generalize the notion of the minimal polynomial to a certain class of transcendental elements of \(\widetilde{\overline{\mathbb Q}}\), where \(\widetilde{\overline{\mathbb Q}}\) denotes the completion of the algebraic closure \({\overline{\mathbb Q}}\) of \({\mathbb Q}\) in \({\mathbb C}\) relative to the spectral norm \(\|x\|= \max\{|\sigma x|\mid \sigma \in G\}\) on \({\overline{\mathbb Q}}\), where \(G= \text{Gal}( {\overline{\mathbb Q}}/{\mathbb Q})\). If \(x\in {\overline{\mathbb Q}}\) is of degree \(n\) over \({\mathbb Q}\), let \(P(z)\) be its minimal polynomial. Then the principal branch of \(F(x,z)= P(z)^{1/n}\) is analytic on any simply connected domain \(D \subseteq {\mathbb C}\cup \{\infty\}\setminus W\) where \(W\) is the set of roots of \(P(z)\), except at \(\infty\) where it has a pole of order \(1\). The objective is to construct such a function, which is called a critical function, for a class of infinite subsets of \({\mathbb C}\) called transcendental divisors. Then, applying this theory, the authors find a function like \(F(x,z)\) for a class of transcendental elements \(x\) of \(\widetilde{\overline{\mathbb Q}}\). These functions are defined on some simply connected domains \(\Omega\subseteq {\mathbb C}\cup \{\infty\}\) such that \(\infty\in \Omega\) and have properties similar of those of the function \(P(z)^{1/n}\).
scientific article; zbMATH DE number 1945291