Proposition

If we use grlex order with $x > y > z$, is $\{ x^4y^2 - z^5, x^3y^3 - 1, x^2y^4 - 2z \}$ a Grobner basis for the ideal generated by these polynomials?

Solution

It is not. Let $g_1, g_2, g_3$ denote the three polynomials. $2xz - y = yg_2 - xg_3 \in I$, thus $2xz \in \LT(I)$.

On the other hand, each $\LT(g_i)$ has a multi-degree of $6$. Therefore, $2xz \notin \{ \LT(g_1), \LT(g_2), \LT(g_3) \}$. Thus $\{ g_1, g_2, g_3 \}$ is not a Grobner basis.