# Introduction

This chapter describes local polynomial rings. Let $R$ be the multivariate polynomial ring $K[x_1, \ldots, x_n]$, where $K$ is a field. We denote by

$$
K[x_1, \ldots, x_n]_{\langle x_1, \ldots, x_n\rangle}
$$

the collection of all rational functions $f/g$ of $x_1, \ldots, x_n$ with $g(p) \not= 0$, where $p=(0, \ldots, 0)$. Such a ring is **local** (has a unique maximal ideal) and we will call it a **local polynomial ring** in Magma. Such a ring is always multivariate and is related to the corresponding multivariate polynomial ring $K[x_1, \ldots, x_n]$ which will we will call **global** (when distinguishing it from the local case). We will also call $K[x_1, \ldots, x_n]_{\langle x_1, \ldots, x_n\rangle}$ the **localization** of $K[x_1, \ldots, x_n]$ (this is always understood to be at the prime ideal generated by $x_1, \ldots, x_n$, corresponding to the origin). Much of the theory for multivariate polynomial rings and their ideals carry over to local polynomial rings, so the reader should first be familiar with multivariate polynomial rings and their ideals (see Chapters [Multivariate Polynomial Rings](../../BasicRings/MultivariatePolynomialRings/index-multivariate-polynomial-rings.md#dpoly-chap) and [Gröbner Bases](../GrobnerBases/index-grobner-bases.md#chapgb)). Corresponding to a Gröbner basis of an ideal of a global multivariate ring is a **standard basis** of an ideal of a local polynomial ring. See [[Cox *et al.*, 1998](../../references.md#cite-uag-clo), Chapter 4] or [[Greuel and Pfister, 2002](../../references.md#cite-grpf02), Chapter 1] for the basics of the theory and algorithms. The other facilities are currently basic but will be expanded in coming versions. But note that computations with $R-$modules, where $R$ is a local polynomial ring, are fully supported: see Chapter [Modules over Multivariate Rings](../ModulesOverMultivariateRings/index-modules-over-multivariate-rings.md#pmod-chap).