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''?
We shall only be interested in orderings on the monomials. In this case, there are some other conditions we shall want to add.
The following example is one of the most commonly used monomial orderings.
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.
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.
where , or, if and . This ordering is called the graded lexicographical ordering or degree ordering, denoted if there is an ambiguiuty.
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 ).