Graphs and recursively defined towers of function fields
A tower of function fields over \(\mathbb F_{q}\) defined recursively by \(f(x,y) \in \mathbb F_{q}[x,y]\) is given by adjoining to \(F_{n-1}\) a root \(x_{n}\) of \(f(x_{n-1},y)\), where \(F_{0} = \mathbb F_{q}(x_{0})\), \(x_{0}\) is an indeterminant, and \(f(x,y)\) is appropriately irreducible. Such towers have been of great use in the literature; here, the author associates to each a directed graph, whose vertices are places of the rational function field of \(\bar \mathbb F_{q}\) and whose directed edges are \(( \alpha, \beta)\) with \(f(\alpha, \beta) = 0\). Interest focuses on sets such as all \(\alpha\) that are the first entry in a directed edge. Under appropriate hypotheses, having in particular to do with ramification, the number of \(\mathbb F_{q}\)-rational places \(Q\) in \(F_{n}\) with \(x_{0}(Q)\) belonging to some directed edge equals the number of paths of length \(n\) in the graph. As stated in the abstract, interesting results are achieved.
scientific article; zbMATH DE number 2162038