  Two fundamental questions in the theory of Groebner bases are decision ("Is a
basis G of a polynomial ideal a Groebner basis?") and transformation ("If it is
not, how do we transform it into a Groebner basis?") This paper considers the
first question. It is well-known that G is a Groebner basis if and only if a
certain set of polynomials (the S-polynomials) satisfy a certain property. In
general there are m(m-1)/2 of these, where m is the number of polynomials in G,
but criteria due to Buchberger and others often allow one to consider a smaller
number.
  This paper presents two original results. The first is a new characterization
theorem for Groebner bases that makes use of a new criterion that extends
Buchberger's Criteria. The second is the identification of a class of
polynomial systems G for which the new criterion has dramatic impact, reducing
the worst-case scenario from m(m-1)/2 S-polynomials to m-1.
