Fonctions analytiques et produits croulants. (Analytic functions and collapsing products)
Let (K,\(| |)\) be a non-archimedean valued complete field that is algebraically closed. Let \(b_ 1,b_ 2,...\in K\) be such that \(\lim_{n\to \infty}| b_ n| =R\), \(| b_ n| <R\) for all n. Let \(\rho >0\) and set \(\Delta:=\{x\in K:| x| <R\), \(| x-b_ n| \geq \rho\) for all \(n\in {\mathbb{N}}\}\). Then there exists a Taylor series g such that \(g(b_ n)=0\) for all n, \(\lim_{| x| \uparrow R, x\in \Delta}| g(x)| =\infty\), and \(1+(1/g)\) is a collapsing product [M. C. Sarmant, Bull. Sci. Math., II. Sér. 109, 155- 178 (1985; Zbl 0564.12024)]. Further, if g is a Taylor series convergent for \(| x| <R\) and \(\lambda\in K\), then g can be factorized as \(\lambda ((1-\pi_{\lambda})/(1-\tau_{\lambda}))\), where \(\pi_{\lambda}\) and \(\tau_{\lambda}\) are collapsing products.
scientific article; zbMATH DE number 3977390