# Multivariate polynomial ring

This is a variation of polynomial ring

View a complete list of variations of polynomial ring OR read a survey article on varying polynomial ring

## Contents

## Definition

Let be a commutative unital ring. The -variate polynomial ring over is defined as the ring of polynomials in symbols. If the symbols are , then the polynomial ring is .

The -variate polynomial ring can be obtained by applying the polynomial ring operator times in succession.

When we simply say multivariate polynomial ring, we usually mean multivariate polynomial ring over a field.

We can also consider the polynomial ring in infinitely many variables over .

## Extra structure

The multivariate polynomial ring over a ring is, first and foremost, a commutative unital ring. However, it has a number of additional structures, as described below.

### As an algebra over the original ring

The polynomial ring naturally gets the structure of a -algebra. In fact it is *free* in the category of -algebras, on generators.

A similar statement holds for polynomial rings in infinitely many variables.

### As a graded ring

The polynomial ring naturally gets the structure of a connected graded -algebra (and hence a graded ring). The graded component is the free -module spanned by all monomials of total degree .

The same holds when we have infinitely many variables.

### As a filtered ring

The polynomial ring naturally gets the structure of a connected filtered -algebra (and hence a filtered ring). The filtered component is the subgroup comprising polynomials of degree at most .