Asymptotic for the number of star operations on one-dimensional Noetherian domains

Abstract

We study the set of star operations on local Noetherian domains D of dimension 1 such that the conductor (D : T ) (where T is the integral closure of D) is equal to the maximal ideal of D. We reduce this problem to the study of a class of closure operations (more precisely, multiplicative operations) in a finite extension k ⊆ B, where k is a field, and then we study how the cardinality of this set of closures vary as the size of k varies while the structure of B remains fixed.