Some question in computing Groebner basis

Question

Problem: Let $G = \{z^5 - z^2, -z^3+y^2,x-1-yz^4\}$ and $I$ be an ideal which is generated by $G$ in $\mathbb{C}[x,y,z]$.

1. Show that $G$ is a Groebner basis of $I$ with lexicographic ordering $x>y>z$.
2. Show that $G$ is not a Groebner basis of $I$ with graded lexicographic ordering $x>y>z$.
3. Find a Groebner basis of $I \cap \mathbb{C}[y,z]$ with lexicographic ordering $y>z$.
4. Find the variety of ideal $I$.
5. Is the polynomial $f = 4xyz^6 - z^3xy - 2z^3y^2 + 2y^4 - yz^4 - 3z^2x^2 + 3xz^2 + x$ in $I$? Explain why?

My attempt:

1. Let $g_1 = z^5 - z^2, g_2 = -z^3+y^2, g_3 = x-1-yz^4$. We compute $$S(g_1,g_2) = y^2(z^5-z^2) - z^5(y^2-z^3) = -y^2z^2 + z^8$$ $$\overline{S(g_1,g_2)}^G = 0$$ $$S(g_1,g_3) = x(z^5-z^2) - z^5(x-yz^4-1) = -xz^2 + yz^9 + z^5$$ $$\overline{S(g_1,g_3)}^G = 0$$ $$S(g_2,g_3) = x(y^2-z^3) - y^2(x-yz^4-1) = -xz^3 + y^3z^4 + y^2$$ $$\overline{S(g_2,g_3)}^G = 0$$ So $G = \{g_1,g_2,g_3\}$ is a Groebner basis of $I$ with the lexicographic ordering.

2. With the graded lexicographic ordering, $g_1=z^5-z^2, g_2=yz^3-y^2, g_3=-yz^4+x-1$ we compute $$S(g_1,g_3) = y(z^5-z^2) + z(-yz^4+x-1) = -yz^2+xz+z$$ $$\overline{S(g_1,g_3)}^G = -yz^2+xz+z \ne 0$$ Hence $G = \{g_1,g_2,g_3\}$ is not a Groebner basis of $I$ with graded lexicographic ordering.

3. We have $G = \{z^5-z^2,y^2-z^3,x-yz^4-1\}$ is a Groebner basis of $I$ with the lexicographic ordering $x>y>z$, so this implies a Groebner basis of $I \cap \mathbb{C}[y,z]$ with the lexicographic ordering $y>z$ is $$G' = \{z^5-z^2,y^2-z^3\}$$

For the last question, I have did it. I have arranged $f$ follows the lexicographic, $f = -3x^2z^2+4xyz^6-xyz^3+3xz^2+x+2y^4-2y^2z^3-yz^4$, since $LT(f) \notin \langle LT(g_1),LT(g_2),LT(g_3) \rangle$, hence $f \notin I$.

I need help in two questions $4$ and $5$. Thank all!

Answer 1

Well, as a note the zero locus of $I$ can be found by restricting the Groebner basis:

$G_2 = G\cap{\Bbb C}[z] = \{z^5-z^2\}$, $G_1 = G\cap{\Bbb C}[y,z] = \{y^2-z^3,z^5-z^2\}$, and $G_1=G$.

First, determine the zero locus of $G_2$ (single complex polynomial in $z$), then extend the zero locus to the zero locus of $G_1$ (taking the zeros you already have for $z$ and determine the corresponding values of $y$), and finally extend the zero locus ot the zero locus of $G$ (similar to the previous step).

More specifically, $z^5-z^2 = z^2(z^3-1)=0$ and the zeros are $0$ (twofold) and the third roots of unity $\xi^0= 1,\xi = e^{2\pi/3},\xi^2$.

Then consider the roots of $y^2-z^3$ using the already computed values for $z$. For $z=0$, one has $y=0$ (twofold) and for $z=1$ one has $y^2=1$, i.e., $y=\pm 1$, for $z=\xi$ one has $y^2=\xi$ and so $y$ is a primitive 6th root of unity, namely $y=e^{2\pi i/6}$ or $y=e^{2\pi i 5/6}$. And so on.

For the last problem, divide $f$ into the Groebner basis. The nice property of Groebner bases is that the residue is zero iff $f\in I$.