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Using polynomial equations to model combinatorial problems has been a populartool both in computational combinatorics as well as an approach to proving newtheorems. In this paper, we look at several combinatorics problems modeled bysystems of polynomial equations satisfying special properties. If the equationsare infeasible, Hilbert's Nullstellensatz gives a certificate of this fact.These certificates have been studied and exhibit combinatorial meaning. In thispaper, we generalize some known results and show that the Nullstellensatzcertificate can be viewed as enumerating combinatorial structures. As such,Gr\"obner basis algorithms for solving these decision problems may implicitlybe solving the enumeration problem as well.

Enumerative Aspects of Nullstellensatz Certificates