Popescu, Angel; Popescu, Nicolae; Zaharescu, Alexandru Transcendental divisors and their critical functions. (English) Zbl 1027.12004 Manuscr. Math. 110, No. 4, 527-541 (2003). 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 \(P(z)^{1/n}\). Reviewer: Gabriel D.Villa-Salvador (México) Cited in 4 ReviewsCited in 3 Documents MSC: 12J05 Normed fields 11R99 Algebraic number theory: global fields 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.) Keywords:minimal polynomials; transcendental elements; analytic continuation PDFBibTeX XMLCite \textit{A. Popescu} et al., Manuscr. Math. 110, No. 4, 527--541 (2003; Zbl 1027.12004) Full Text: DOI