Monomial order

In mathematics, a monomial order (sometimes called a term order or an admissible order) is a total order on the set of all (monic) monomials in a given polynomial ring, satisfying the property of respecting multiplication, i.e.,

Monomial orderings are most commonly used with Gröbner bases and multivariate division. In particular, the property of being a Gröbner basis is always relative to a specific monomial order.

Besides respecting multiplication, monomial orders are often required to be well-orders, since this ensures the multivariate division procedure will terminate. There are however practical applications also for multiplication-respecting order relations on the set of monomials that are not well-orders.

In the case of finitely many variables, well-ordering of a monomial order is equivalent to the conjunction of the following two conditions:

Since these conditions may be easier to verify for a monomial order defined through an explicit rule, than to directly prove it is a well-ordering, they are sometimes preferred in definitions of monomial order.

The choice of a total order on the monomials allows sorting the terms of a polynomial. The leading term of a polynomial is thus the term of the largest monomial (for the chosen monomial ordering.

Concretely, let for be any ring of polynomials $R$ , the set $M$ of the (monic) monomials in $R$ is a basis of $R$ , considered as a vector space over the field of the coefficients. Thus, any nonzero polynomial $p$ in $R$ has a unique expression $p=\textstyle \sum _{u\in S}c_{u}u$ as a linear combination of monomials, where $S$ is a finite subset of $M$ and the $c u$ are all nonzero. When a monomial order has been chosen, the leading monomial is the largest $u$ in $S$ , the leading coefficient is the corresponding $c u$ , and the leading term is the corresponding $c u u$ . Head monomial/coefficient/term is sometimes used as a synonym of "leading". Some authors use "monomial" instead of "term" and "power product" instead of "monomial". In this article, a monomial is assumed to not include a coefficient.

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