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:
- If u < v and w is any other monomial, then uw
. In other words, the ordering respects multiplication. - 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<x
Monomial orderings are most commonly used with Gröbner bases and multivariate division.
Read more about Monomial Order: Examples, Related Notions
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