Monomial Order

In mathematics, a monomial order is a total order on the set of all (monic) monomials in a given polynomial ring, satisfying the following two properties:

  1. If u < v and w is any other monomial, then uw. In other words, the ordering respects multiplication.
  2. The ordering is a well ordering (every non-empty set of monomials has a minimal element).

Among the powers of any one variable x, the only ordering satisfying these conditions is the natural ordering 1<xx would fail to have a minimal element). Therefore the notion of monomial ordering is interesting only in the case of multiple variables.

Monomial orderings are most commonly used with Gröbner bases and multivariate division.

Read more about Monomial Order:  Examples, Related Notions

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