Reduced Grobner Basis With Double Exponential Cardinality

In this article, we investigate the cardinality of Grobner basis under various orderings. We identify a family of polynomials F and a criteria for the monomial orderings such that the reduced Grobner basis is double exponential in cardinality. We also show that the said criteria is satisfied by orderings such as lexicographic, degree lexicographic and weighted orderings.