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
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 .
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 .