Divisors, measures and critical functions (Q732862)

In a previous work [Math. Sci. Res. J. 8, No. 1, 1--15 (2004; Zbl 1088.11084)], the authors introduced a new distance between Galois orbits over \({\mathbb Q}\). If \(\bar{\mathbb Q}\) denotes a fixed algebraic closure of \({\mathbb Q}\) contained in \({\mathbb C}\), the usual trace map \(\text{Tr}: (\bar{\mathbb Q},|\cdot|)\to ({\mathbb C},|\cdot|)\) is not continuous with the usual absolute value. However, \(\text{Tr}: (\bar{\mathbb Q},\|\cdot\|_s)\to ({\mathbb C}, |\cdot|)\) is continuous where \(\|\cdot\|_s\) is the spectral norm defined by \[ \|x\|_s:=\max\{\sigma(x)\mid \sigma\in \roman{Gal}(\bar{\mathbb Q}/ {\mathbb Q})\} \] since \(|\text{Tr}\,\alpha-\text{Tr}\,\beta|\leq \|\alpha-\beta\|\}_s\). In this paper the authors continue the work started in the paper cited above and generalize results of \textit{A. Popescu, N. Popescu} and \textit{A. Zaharescu} [Manuscr. Math. 110, No. 4, 527--541 (2003; Zbl 1027.12004)]. The main goal is to generalize the critical function \(F(\alpha,z)=[P_{\alpha}(z)]^{1/n}\) of an algebraic number to a class of transcendental numbers. Here \(P_{\alpha}\) is the minimal polynomial of \(\alpha\). For \(\alpha\in \bar{\mathbb Q}\), let \(\alpha_1=\alpha,\alpha_2,\dots,\alpha_n\) be the conjugates of \(\alpha\). To \(\alpha\) is associated a positive unitary divisor \[ \psi(\alpha)=\Big(\alpha_1,\alpha_2,\dots,\alpha_n,\tfrac{1}{n}, \tfrac{1}{n},\dots, \tfrac{1}{n}\Big). \] The paper is organized as follows. On the set \(D\) of positive unitary divisors, the authors have introduced a distance \(d\). Some properties of \((D, d)\) studied in their previous work are recalled in Section 2. In Section 3 they construct a sequence of algebraic numbers \((\alpha_n)_{ n\in{\mathbb N}}\) that does not converge in the spectral norm while the corresponding \((\psi(\alpha_n))_{n\in {\mathbb N}}\) converges in the completion of \(D\) with respect to \(d\). In Section 4 some properties of the trace of unitary divisors found in their previous paper are extended to Lipschitzian functions on bounded domains in \({\mathbb C}\) using a Dirac measure. In Section 5, the authors define the critical function associated to a divisor with compact support. It is shown that given a generalized divisor \(W\) in the completion of \(D\) with compact support and a sequence \((W_n)_{n\in{\mathbb N}}\) in \(D\), \(W_n\to W\) with respect \(d\), the sequence of analytic functions \(z\to F(W_n,z)-z\) converges uniformly on compact subsets of a set \(\Omega\) to an analytic function \(z\to F(W,z)-z\) where \(\infty\in \Omega\subseteq ({\mathbb C}\cup \{\infty\})\setminus {\mathcal K}\), \({\mathcal K}\) compact.