Subring -- The type of all subrings

Description

Subring is a type that stores information associated to a subring of a polynomial ring or quotient ring, such as a set of subring generators and a reference to the ring containing these generators. An instance of a Subring is constructed with the function subring. For many uses, it is suggested to use a Subring, as the computation objects (SAGBIBasis) are handled behind the scenes, and the user experience is more streamlined.

Every instance of Subring has the following keys (some of which are strings):

ambientRing: The polynomial or quotient ring that contains the subring instance's generators.
generators: A one-row matrix, the generators of the subring.
cache: Contains data from previous computations to allow for more efficient computations.
presentationRing: the polynomial ring with one variable for each generator of the subring.
presentationMap: a map from the presentation ring to the ambient ring of the subring.

Types of subring:

IntersectedSubring -- The type of all subrings arising from intersection

Methods that use a subring:

ambient(Subring) -- The ambient ring of a subring
flattenedRing(Subring) -- see flattenedRing -- The flattened ring of a subring or sagbiBasis
forceSB(Subring) -- declare the generators to be a complete subalgebra basis
generators(Subring) -- A generating set of a subring
groebnerMembershipTest(RingElement,Subring) -- see groebnerMembershipTest -- Extrinsic method for subring membership
groebnerSubductionQuotient(RingElement,Subring) -- see groebnerSubductionQuotient -- Extrinsic method for subduction quotients
intersect(Subring,Subring) -- Intersection of subrings
isSAGBI(Subring) -- Check if the generators are a subalgebra basis
net(Subring) -- Short summary of a subring
numgens(Subring) -- The number of generators of a subring
presentationRing(Subring) -- see presentationRing -- returns the presentation ring of a subring
Matrix % Subring -- see Reduction in subrings -- Remainder modulo a subring
RingElement % Subring -- see Reduction in subrings -- Remainder modulo a subring
RingElement // Subring -- Subduction quotient with respect to a subring
sagbi(Subring) -- see sagbi -- Compute a subalgebra basis (sagbi basis)
sagbiBasis(Subring) -- see sagbiBasis -- Constructs a computation object from a subring.
subalgebraBasis(Subring) -- see subalgebraBasis -- Compute subalgebra basis generators
subduction(Subring,Matrix) -- see subduction -- Subduction against a set elements
subduction(Subring,RingElement) -- see subduction -- Subduction against a set elements

For the programmer

The object Subring is a type, with ancestor classes HashTable < Thing.

The source of this document is in SubalgebraBases/documentation.m2:1616:0.

Subring -- The type of all subrings

Description

See also

Types of subring:

Methods that use a subring:

For the programmer