Results on the Ratliff-Rush Closure and the Integral Closedness of Powers of Certain Monomial Curves

TL;DR

This paper computes Groebner bases for monomial curve ideals from arithmetic sequences, introduces a method to generate Ratliff-Rush ideals in multivariable polynomial rings, and characterizes when their powers are integrally closed.

Contribution

It presents a new procedure for generating infinite families of Ratliff-Rush ideals from two-variable cases and proves all powers of initial ideals are Ratliff-Rush, extending previous work.

Findings

All powers of inP are Ratliff-Rush ideals.

Provides necessary and sufficient conditions for integral closedness of powers.

Generalizes previous results for arithmetic sequences.

Abstract

Starting from \cite{Ayy2} we compute the Groebner basis for the defining ideal, P, of the monomial curves that correspond to arithmetic sequences, and then give an elegant description of the generators of powers of the initial ideal of P, inP. The first result of this paper introduces a procedure for generating infinite families of Ratliff-Rush ideals, in polynomial rings with multivariables, from a Ratliff-Rush ideal in polynomial rings with two variables. The second result is to prove that all powers of inP are Ratliff-Rush. The proof is through applying the first result of this paper combined with Corollary (12) in \cite{Ayy4}. This generalizes the work of \cite{Ayy1} (or \cite{Ayy11}) for the case of arithmetic sequences. Finally, we apply the main result of \cite{Ayy3} to give the necessary and sufficient conditions for the integral closedness of any power of inP.