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.

翻译：本文研究了Gröbner基在不同单项式序下的规模问题。我们构造了一族多项式F并给出一个判定准则，使得对于满足该准则的任意单项式序，F的约化Gröbner基均具有双重指数规模的基数。同时证明了该准则在诸如字典序、度字典序及加权序等常见序下均成立。