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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“My trade and my art is living. He who forbids me to speak about it according to my sense, experience, and practice, let him order the architect to speak of buildings not according to himself but according to his neighbor; according to another mans knowledge, not according to his own.”
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