On the monomial reduction number of a monomial ideal in $K[x,y]$

TL;DR

This paper investigates the reduction number of monomial ideals in two-variable polynomial rings, providing methods to compute it via linear inequalities and explicit formulas in special cases.

Contribution

It introduces a systematic approach to compute the reduction number of monomial ideals in $K[x,y]$, including explicit formulas for certain cases.

Findings

Reduction number can be computed through linear inequalities.

Explicit formulas derived for special classes of monomial ideals.

Provides a framework for analyzing reduction numbers in two-variable polynomial rings.

Abstract

The reduction number of monomial ideals in the polynomial $K[x,y]$ is studied. We focus on ideals $I$ for which $J=(x^a,y^b)$ is a reduction ideal. The computation of the reduction number amounts to solve linear inequalities. In some special cases the reduction number can be explicitly computed.