We will see in the next sections that with a lex order basis it is easy to count the solutions explicitly (if there are a finite number). But how do we tell there are a finite number?

There are a finite number of solutions if and only if for each variable
there is a term **t** in the leading monomials of the Gröbner basis such that
, that is, that at least one of the members of the Gröbner basis
has its leading term as a pure power of .

This gives us a simple and rational test for finite cardinality of solutions, given some Gröbner basis. Any monomial ordering may be used.

** Example.** If , then the total
degree (graded reverse lexicographic or grevlex) Gröbner basis contains the
polynomials and .
The reader should verify that the terms and
are the highest-multidegree
terms in these polynomials. Since they are pure powers, and the condition is
satisfied for each variable, there are only a finite number of solutions.

Tue Mar 12 21:09:19 EST 1996