Reduced Groebner Bases With Double Exponential Cardinality

In this article, we investigate the cardinality of Groebner bases under various monomial orderings. We identify a family of polynomials F and a criterion such that the reduced Groebner basis of F is double exponential in cardinality with respect to any monomial ordering which satisfies this criterion. We also show that the said criterion is satisfied by orderings such as the lexicographic, degree lexicographic and weighted orderings.