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 and is any other monomial, then . 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: The order is a total order. If u is any monomial then . 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 R be any ring of polynomials. Then 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 as a linear combination of monomials, where S is a finite subset of M and the cu 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 cu, and the leading term is the corresponding cuu. 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".