Primary Decomposition of Powers of the Prime Ideal of a Numerical Semigroup Ring

Let R=k[tn1,…,tns]=k[x1,…,xs]/P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R=k[t^{n_{1}},\ldots ,t^{n_{s}}]=k[x_{1},\ldots ,x_{s}]/P$\end{document} be a numerical semigroup ring and let P(n) = PnRP ∩ R be the symbolic power of P and Rs(P) = ⊕i≥ 0P(n)tn the symbolic Rees ring of P. It is hard to determine symbolic powers of P; there are even non-Noetherian symbolic Rees rings for 3-generated semigroups. We determine the primary decomposition of powers of P for some classes of 3-generated numerical semigroups.