Star Operations on Kunz Domains

Abstract

We study star operations on Kunz domains, a class of analytically irreducible, residually rational domains associated to pseudo-symmetric numerical semigroups, and we use them to refute a conjecture of Houston, Mimouni and Park. We also find an estimate for the number of star operations in a particular case, and a precise counting in a sub-case.

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

“…They analyzed in particular the Noetherian case, showing that |Star(D)| = ∞ when D has dimension greater than 1 [8,Theorem 2.1], showing how to reduce to the local case [8,Theorem 2.3] and calculating the cardinality of Star(D) when ℓ D (T /m D ) ≤ 3, where T is the integral closure of D and m D the maximal ideal of D [8, Theorem 3.1]. Further cases have been considered, for example, in [8] (for infinite residue field), in [15,18] (for pseudo-valuation domains), in [17] (for Kunz domains) and [22] (for some numerical semigroup rings).…”