# GroebnerBasis -- the class of all Gröbner bases

## Description

A Gröbner basis in Macaulay2 consists of a Gröbner basis computation, and several associated matrices. Normally you don't need to refer to these objects directly, as many operations on matrices and modules create them, and refer to them. For more information, see Gröbner bases.

## Functions and methods returning a Gröbner basis :

• forceGB -- declare that the columns of a matrix are a Gröbner basis
• "forceGB(Matrix)" -- see forceGB -- declare that the columns of a matrix are a Gröbner basis
• gb -- compute a Gröbner basis
• "gb(Ideal)" -- see gb -- compute a Gröbner basis
• "gb(Matrix)" -- see gb -- compute a Gröbner basis
• "gb(Module)" -- see gb -- compute a Gröbner basis
• "markedGB" -- see markedGB(Matrix,Matrix) -- make a marked Gröbner basis
• markedGB(Matrix,Matrix) -- make a marked Gröbner basis

## Methods that use a Gröbner basis :

• "Number % GroebnerBasis" -- see % -- a binary operator, usually used for remainder and reduction
• "betti(GroebnerBasis)" -- see betti -- display or modify a Betti diagram
• generators(GroebnerBasis) -- the generator matrix of a Gröbner basis
• "getChangeMatrix(GroebnerBasis)" -- see getChangeMatrix -- get the change of basis matrix
• "leadTerm(GroebnerBasis)" -- see leadTerm(Matrix) -- get the greatest term of each column
• Matrix % GroebnerBasis -- calculate the normal form of ring elements and matrices using a (partially computed) Gröbner basis
• "RingElement % GroebnerBasis" -- see Matrix % GroebnerBasis -- calculate the normal form of ring elements and matrices using a (partially computed) Gröbner basis
• "Matrix // GroebnerBasis" -- see Matrix // Matrix -- factor a map through another
• "RingElement // GroebnerBasis" -- see Matrix // Matrix -- factor a map through another
• mingens(GroebnerBasis) -- (partially constructed) minimal generator matrix
• quotient(Matrix,GroebnerBasis) -- matrix quotient
• "quotientRemainder(Matrix,GroebnerBasis)" -- see quotientRemainder -- matrix quotient and remainder
• "remainder(Matrix,GroebnerBasis)" -- see remainder -- matrix remainder
• "ring(GroebnerBasis)" -- see ring -- get the associated ring of an object
• "status(GroebnerBasis)" -- see status -- status of a resolution computation
• syz(GroebnerBasis) -- retrieve the syzygy matrix
• target(GroebnerBasis) -- find target of a Gröbner basis

## For the programmer

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