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Recipe 2---existence of a finite number of solutions

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.

Robert Corless
Tue Mar 12 21:09:19 EST 1996