# EuclideanGroebnerBasisPackage(Dom, Expon, VarSet, Dpol)¶

- Dom: EuclideanDomain
- Expon: OrderedAbelianMonoidSup
- VarSet: OrderedSet
- Dpol: PolynomialCategory(Dom, Expon, VarSet)

EuclideanGroebnerBasisPackage computes groebner bases for polynomial ideals over euclidean domains. The basic computation provides a distinguished set of generators for these ideals. This basis allows an easy test for membership: the operation euclideanNormalForm returns zero on ideal members. The string “info” and “redcrit” can be given as additional args to provide incremental information during the computation. If “info” is given, a computational summary is given for each `s`

-polynomial. If “redcrit” is given, the reduced critical pairs are printed. The term ordering is determined by the polynomial type used. Suggested types include DistributedMultivariatePolynomial, HomogeneousDistributedMultivariatePolynomial, GeneralDistributedMultivariatePolynomial.

- euclideanGroebner: (List Dpol, String) -> List Dpol
`euclideanGroebner(lp, infoflag)`

computes a groebner basis for a polynomial ideal over a euclidean domain generated by the list of polynomials`lp`

. During computation, additional information is printed out if infoflag is given as either “info” (for summary information) or “redcrit” (for reduced critical pairs)

- euclideanGroebner: (List Dpol, String, String) -> List Dpol
`euclideanGroebner(lp, "info", "redcrit")`

computes a groebner basis for a polynomial ideal generated by the list of polynomials`lp`

. If the second argument is`"info"`

, a summary is given of the critical pairs. If the third argument is “redcrit”, critical pairs are printed.

- euclideanGroebner: List Dpol -> List Dpol
`euclideanGroebner(lp)`

computes a groebner basis for a polynomial ideal over a euclidean domain generated by the list of polynomials`lp`

.

- euclideanNormalForm: (Dpol, List Dpol) -> Dpol
`euclideanNormalForm(poly, gb)`

reduces the polynomial`poly`

modulo the precomputed groebner basis`gb`

giving a canonical representative of the residue class.