# Size of the Groebner basis and change of coordinates

Given an ideal $I$ of $k[x,y]$, the size of the Groebner basis depends on the monomial ordering. For example, if $$I = \langle x^3y^4 , x^2 + y^2 \rangle,$$ then the Groebner basis with the lexicographical order $x,y$ is $$x^2 + y^2, xy^6, y^8,$$ and it has 3 elements.
If we choose the lexicographical order $y,x$, the Groebner basis is $$x^2 + y^2, x^7,$$ that has 2 elements. Equivalently, we can think of the change of coordinates $x, y \to y, x$ and we get that the ideal $$I' = \langle y^3x^4 , x^2 + y^2 \rangle,$$ with Groebner basis of 2 elements (with the lexicographical order $x,y$). More generally: a change of coordinates $x, y \to x', y'$ changes the cardinality of the Groebner basis.

Now, let $I$ be an ideal in $k[x_1,\ldots,x_n]$, such that the Groebner basis with respect to the monomial lexicographical order $x_1,\ldots,x_n$ has $m$ elements. Let $x_1,\ldots,x_n \to x'_1,\ldots,x'_n$ be a change of coordinates and $I'$ the ideal that we get by changing coordinates on $I$ (replacing $x_i$ by $x'_i$).

The question is: does there exist a permutation $\sigma$ such that the Groebner basis of $I'$ with respect to the monomial lexicographical order $x_{\sigma_1},\ldots,x_{\sigma_n}$ contains exactly $m$ elements?