Simple generators with a complex Gröbner basis

Question

It's known that finding a Gröbner basis of a polynomial ideal has a worst-case space complexity of $O(2^{2^{c\cdot n}})$, where c is constant and n is the number of variables $k[x_1,\ldots,x_n]$.

However, in practice it seems that most ideals have a simple Gröbner basis.

Can anyone give some concrete examples of small generators whose ideal has a large Gröbner basis? How would I go about searching for such examples (besides a brute-force approach of trying random ideals)?

Answer 1

I just found the following example, from this paper:

GroebnerBasis[{x^5 + y^5 + z^5 - 1, x^3 + y^3 + z^2 - 1}, {x, y, z}] (* Is long *)