Analytic mappings in the tree \(\text{Mult}(K[x])\) (Q2574778)

Let \(K\) be an algebraically closed complete ultrametric field, let \(D \subset K\) be a closed bounded set. This paper is the natural continuation of previous ones of the authors devoted to the study of the Banach \(K\)-algebra \(H(D)\) of analytic elements on \(D\), which is equipped with the \(K\)-algebra norm \(\| \; . \; \| _D\) of uniform convergence on \(D\). The set of all continuous multiplicative seminorms on \(H(D)\) is denoted by \(\text{Mult}(H(D),\| \; . \; \| _D)\). It is a subset of \(\text{Mult}(K[x])\), the set of all multiplicative seminorms on \(K[x]\). The elements of \(\text{Mult}(K[x])\) and of \(\text{Mult}(H(D),\| \; . \; \| _D)\) can be characterized in terms of certain filters on \(K\), the so-called circular filters. Then \(\text{Mult}(K[x])\) is provided with two topologies: the usual topology of simple convergence for which it is locally compact, and a metric topology. This last one, which is strictly stronger than the topology of simple convergence, comes from a partial order that gives \(\text{Mult}(K[x])\) a tree structure for which it is complete. The main purpose of the authors is to study maps from \(\text{Mult}(H(D),\| \; . \; \| _D)\) to \(\text{Mult}(K[x])\) that are defined by elements of \(H(D)\). They show that such maps are continuous for the two topologies on \(\text{Mult}(K[x])\), and uniformly continuous for the metric topology. They also give a description, in terms of Shilov boundaries, of the boundary of (\(\text{Mult}(H(D),\| \; . \; \| _D)\) inside \(\text{Mult}(K[x])\) with respect to the two topologies on it. Further, at the end of the paper the authors present some results related to the Kranser-Tate algebra \(H(D) = K \{ h \}[x]\) (\(h \in K(x)\) of the form \(P(x)/Q(x)\), \(P, Q \in K[x]\), \(\deg(P) > \deg(Q)\), and \(D = h^{-1}(d(0,1))\) where \(d(0,1)\) is the unit disk in \(K\)). Among them we point out the result assuring that the Gauss norm on \( K \{ t \}\) admits a number of extensions to \( K \{ t \}[x]\) which is precisely equal to the cardinal number of the Shilov boundary of \(\text{Mult}(H(D),\| \; . \; \| _D)\).