A fast algorithm for constructing Arf closure and a conjecture (Q404562)

Let \(R=k[[x_1(t), \ldots, x_n (t)]]\subseteq k[[t]]\) such that \(\dim_k k[[t]]/R<\infty\). Let \(\Gamma(R)\) be the semi--group of \(R\). For any \(n\in \Gamma(R)\) choose \(S_n\in R\) such that \(\text{ord}(S_n)=n\). \(R\) is called Arf ring if \(\{rS_n^{-1}|\text{ord} (r)\geq n\}\) is a ring for alle \(n\in \Gamma(R)\). The Arf closure of \(R\) is the smallest Arf ring containing \(R\).NEWLINENEWLINEAn algorithm is given for computing the Arf closure. The relation between rings with the same Arf closure is studied.