# Mathieu-Zhao spaces of polynomial rings

## TL;DR

This paper characterizes all Mathieu-Zhao spaces within polynomial rings over algebraically closed fields of characteristic zero that contain finite codimension ideals, and provides an algorithm to identify specific Mathieu-Zhao spaces.

## Contribution

It offers a complete classification of Mathieu-Zhao spaces containing finite codimension ideals and introduces an algorithm for their identification.

## Key findings

- All Mathieu-Zhao spaces containing finite codimension ideals are described.
- An explicit algorithm is provided to determine if a subspace is a Mathieu-Zhao space.
- The results apply to polynomial rings over algebraically closed fields of characteristic zero.

## Abstract

We describe all Mathieu-Zhao spaces of $k[x_1,\cdots,x_n]$ ($k$ is an algebraically closed field of characteristic zero) which contains an ideal of finite codimension. Furthermore we give an algorithm to decide if a subspace of the form $I+kv_1+\cdots+kv_r$ is a Mathieu-Zhao space, in case the ideal $I$ has finite codimension.