We need two more definitions (Definition 4, p. 82 of Cox, Little, and O'Shea).

- If multideg and multideg, then let
where for each
**i**. We call the**least common multiple**of LM and LM. We write . - The
**S-polynomial**of**f**and**g**is the combination(Note that we are inverting the leading coefficients here as well.)

For example, let and with
the `tdeg`

(graded reverse lexicographic) order. Then
and

The idea of an S-polynomial is that * leading terms are cancelled*.
It turns out that every kind of cancellation can be expressed as a sequence
of these S-polynomial combinations.

** Definition of a Gröbner Basis** (Theorem 6 in Cox, Little, and O'Shea,
p. 84.)

Let **I** be a polynomial ideal. Then a basis
for **I** is a Gröbner basis for **I** if and only if for all pairs ,
the remainder on division of by **G** (listed in some order)
is zero.

This is a computationally verifiable definition---to test if a given
basis is a Gröbner basis, form all S-polynomials,
and compute the remainders modulo **G**. If any of these remainders are
nonzero, it isn't a Gröbner basis.

Tue Mar 12 21:09:19 EST 1996