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.,

If

u

≤

v

{\displaystyle u\leq v}

and

w

{\displaystyle w}

is any other monomial, then

u

w

≤

v

w

{\displaystyle uw\leq vw}

.

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.

Definition, details and variations

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:

The order is a total order.

If u is any monomial then

1

≤

u

{\displaystyle 1\leq u}

.

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.

Leading monomials, terms, and coefficients

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).