To say what a leading term'' is, we need to be able to order the monomials in some way. The amounts to the same thing as ordering the set . What is an ordering''?

Definition 2.8.1   Let be a set and let be a function 2.9. We call a total ordering 2.10 if
• , (reflective'')
• if and then , (anti-symmetric'')
• if and then , (transitive'')
• for all , either or .
A function satisfying only the first three conditions above is called a partial ordering.

We shall only be interested in orderings on the monomials. In this case, there are some other conditions we shall want to add.

Definition 2.8.2   Let be the set of monomials in the variables . A relation on is called a monomial ordering if, as an ordering on it satisfies
• is a total ordering on ,
• if and then ,
• is a well-ordering 2.11 on .

The following example is one of the most commonly used monomial orderings.

Example 2.8.3   Define . In general, we say if the first non-zero entry in is positive.

Another perspective: Think of as the letter a'', as the word aa'', ..., as the letter b'', as the word ab'', ... .In general, we think of as the word having a''s, followed by b''s, ... . We say if the word for occurs before the word for in the dictionary.

This ordering is called the lexicographical ordering, denoted if there is an ambiguiuty.

Lemma 2.8.4   The lexicographical ordering is a monomial ordering.

The proof is omitted (see Cox, Little, O'Shea [CLO], Proposition 4, Chapter 2, §2, for example).

The following example is another one of the most commonly used monomial orderings.

Example 2.8.5   In general, we say if either

where , or, if and . This ordering is called the graded lexicographical ordering or degree ordering, denoted if there is an ambiguiuty.

Lemma 2.8.6   The graded lexicographical ordering is a monomial ordering.

This proof is also omitted.

Let be a fixed monomial ordering on . We define leading term of (with respect to ) to be the largest of the monomial terms occurring in the expression for with respect to . Since the leading term is used so often, we introduce a short-hand notation for it. For , let denote the leading term of (with respect to ).

david joyner 2008-04-20