# PolynomialRing -- the class of all ordered monoid rings

## Description

Every element of a polynomial ring is also a RingElement.

## Functions and methods returning a polynomial ring :

• degreesRing -- the ring or monoid of degrees
• "InexactFieldFamily Array" -- see Ring Array -- the standard way to make a polynomial ring
• Ring Array -- the standard way to make a polynomial ring
• "InexactFieldFamily List" -- see Ring List -- make a local polynomial ring
• Ring List -- make a local polynomial ring
• "InexactFieldFamily Monoid" -- see Ring Monoid -- make a polynomial ring
• Ring Monoid -- make a polynomial ring

## Methods that use a polynomial ring :

• "ambient(PolynomialRing)" -- see ambient(Ring) -- ambient polynomial ring
• "codim(PolynomialRing)" -- see codim(QuotientRing) -- compute the codimension
• degreeGroup(PolynomialRing) (missing documentation)
• "describe(PolynomialRing)" -- see describe -- real description
• "dim(PolynomialRing)" -- see dim(Ring) -- compute the Krull dimension
• "flattenRing(PolynomialRing)" -- see flattenRing -- write a ring as a (quotient of a) polynomial ring
• "Grassmannian(ZZ,ZZ,PolynomialRing)" -- see Grassmannian(ZZ,ZZ) -- the Grassmannian of linear subspaces of a vector space
• "heft(PolynomialRing)" -- see heft -- heft vector of ring or monoid
• hilbertSeries(PolynomialRing) -- compute the Hilbert series of a ring
• "isAffineRing(PolynomialRing)" -- see isAffineRing -- whether something is an affine ring
• isHomogeneous(PolynomialRing) (missing documentation)
• "isPolynomialRing(PolynomialRing)" -- see isPolynomialRing -- whether something is a polynomial ring
• "isSkewCommutative(PolynomialRing)" -- see isSkewCommutative -- whether a ring has skew commuting variables
• isWeylAlgebra(PolynomialRing) (missing documentation)
• "newCoordinateSystem(PolynomialRing,Matrix)" -- see newCoordinateSystem -- change variables
• "newRing(PolynomialRing)" -- see newRing -- make a copy of a ring, with some features changed
• "numgens(PolynomialRing)" -- see numgens(Ring) -- number of generators of a polynomial ring
• "options(PolynomialRing)" -- see options(Monoid) -- get values used for optional arguments
• "precision(PolynomialRing)" -- see precision
• presentation(PolynomialRing,QuotientRing) -- presentation of a quotient ring
• selectVariables(List,PolynomialRing) -- make a subring of a polynomial ring generated by selected variables

## For the programmer

The object PolynomialRing is a type, with ancestor classes EngineRing < Ring < Type < MutableHashTable < HashTable < Thing.