Gröbner bases

The groebner package contains some functions dealing with ideals of multivariate polynomial rings over a field. In particular, Gröbner bases of such ideals can be computed.

An ideal is given by a list of generators. They must all be polynomials of the same type, i.e., for all of them, the coefficient ring (third operand) and list of unknowns (second operand) must be the same. The generators may also be expressions (all of them must be, if any of them is).

Gröbner bases and related notions depend on the monomial ordering (also called term ordering) under consideration. MuPAD® knows the following orderings:

  • the lexicographical ordering, denoted by the identifier LexOrder.

  • the ordering by total degree, with the lexicographical ordering used as a tie-break; it is denoted by the identifier DegreeOrder.

  • the ordering by total degree, with the opposite of the lexicographical ordering for the reverse order of unknowns used as a tie-break (i.e., the monomial that is lexicographically smaller if the order of variables is reversed, is considered the bigger one); this one is denoted by DegInvLexOrder.

  • user-defined orderings. They constitute a domain Dom::MonomOrdering of their own.

Orderings always refer to the order of the unknowns of the polynomial; e.g., x is lexicographically bigger than y in F[x, y], but smaller than y when regarded as an element of F[y, x].

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