Let \(g_1 = x^2-xy\), \(g_2 = xy-1\), and \(G_0 = (g_1, g_2)\). We compute \(S(g_1,g_2)=y(x^2-xy)-x(xy-1)=-xy^2+x\). Set \(g_3 = \overline{-xy^2+x}^{G_0}=x-y\). Let \(G_1 = (g_1,g_2,g_3)\).

Now, we compute:

• \(S(g_1,g_2)^{G_1}=\overline{-xy^2+x}^{G_1}=0\)
• \(S(g_1,g_3)^{G_1}=\overline{0}^{G_1}=0\)
• \(S(g_2,g_3)^{G_1}=\overline{y^2 - 1}^{G_1}= y^2 - 1\)

Set \(g_4=y^2-1\) and set \(G_2= (g_1,g_2,g_3,g_4)\). We know in advance that \(S(g_1,g_2)^{G_2}=S(g_1,g_3)^{G_2}=S(g_2,g_3)^{G_2}=0\) since this is the purpose of \(g_4\). We compute:

• \(S(g_1,g_4)^{G_2}=\overline{x^2 - xy^3}^{G_2}=0\)
• \(S(g_2,g_4)^{G_2}=\overline{x-y}^{G_2}=0\)
• \(S(g_3,g_4)^{G_2}=\overline{x - y^3}^{G_2}=0\)

Thus a Gröbner basis is \[\{ x^2-xy,\ xy-1,\ x-y,\ y^2-1 \}.\]

We use Sage:

sage: R.<x,y,z> = PolynomialRing(QQ, order="lex")
sage: ideal([x^2*y-1, x*y^2-x]).groebner_basis()
[x^2 - y, y^2 - 1]
sage: ideal([x^2+y, x^4+2*x^2*y+y^2+3]).groebner_basis()
[1]
sage: ideal([x-z^4, y-z^5]).groebner_basis()
[x - z^4, y - z^5]
sage: R.<x,y,z> = PolynomialRing(QQ, order="deglex")
sage: ideal([x^2*y-1, x*y^2-x]).groebner_basis()
[x^2 - y, y^2 - 1]
sage: ideal([x^2+y, x^4+2*x^2*y+y^2+3]).groebner_basis()
[1]
sage: ideal([x-z^4, y-z^5]).groebner_basis()
[x^4 - y^3*z, y^2*z^2 - x^3, y*z^3 - x^2, z^4 - x, x*z - y]

Thus our Groebner bases are as follows:

1. • lex: \(\{ x^2 - y,\ y^2 - 1 \}\)
   • grlex: \(\{ x^2 - y,\ y^2 - 1 \}\)
2. • lex: \(\{ 1 \}\)
   • grlex: \(\{ 1 \}\)
3. • lex: \(\{ x - z^4,\ y - z^5 \}\)
   • grlex: \(\{ x^4 - y^3z,\ y^2z^2 - x^3,\ yz^3 - x^2,\ z^4 - x,\ xz - y \}\)

We use Sage:

sage: R.<x,y> = PolynomialRing(QQ, order="lex")
sage: I = ideal([-x^3+y, x^2*y-z])
sage: f = x*y^3-z^2+y^5-z^3
sage: G = I.groebner_basis()
sage: G
[y^5 - z^3, x*y^3 - z^2, x^3 - y, x^2*y - z, x*z - y^2]
sage: division_algorithm(f,G)
([1, 1, 0, 0, 0], 0)

Since the element \(f\) has remainder \(0\), it is contained in the ideal \(I\).

We use Sage:

sage: R.<x,y,z> = PolynomialRing(QQ,order="lex")
sage: I = ideal([x^2+y^2+z^2-1, x^2+y^2+z^2-2*x, 2*x-3*y-z])
sage: I.groebner_basis()
[x - 1/2, y + 1/3*z - 1/3, z^2 - 1/5*z - 23/40]

We solve to find \(z = \frac{1}{10} \pm \frac{3}{20} \sqrt{26}\) and then our two solutions are \(\left(\frac{1}{2},\ \frac{3}{10} + \frac{1}{20} \sqrt{26},\ \frac{1}{10} + \frac{3}{20} \sqrt{26} \right)\) and \(\left(\frac{1}{2},\ \frac{3}{10} - \frac{1}{20} \sqrt{26},\ \frac{1}{10} - \frac{3}{20} \sqrt{26} \right)\)

We set \(a=\cos(t), b= \sin(t), c=\cos(u), d=\sin(u)\) and note that these must satisfy \(a^2+b^2=1\) and \(c^2+d^2=1\). Thus we want to compute an elimination ideal as follows:

sage: R.<a,b,c,d,x,y,z> = PolynomialRing(QQ,order="lex")
sage: I = ideal([x-(2+a)*c, y-(2+a)*d, z-b, a^2+b^2-1, c^2+d^2-1])
sage: I.groebner_basis()
[a - 1/4*x^2 - 1/4*y^2 - 1/4*z^2 + 5/4, b - z, c^2 + d^2 - 1,
    c*x + d*y - 1/4*x^2 - 1/4*y^2 - 1/4*z^2 - 3/4, c*y - d*x,
    c*z^2 + 3*c + 1/4*x^3 + 1/4*x*y^2 + 1/4*x*z^2 - 13/4*x,
    d*x^2 + d*y^2 - 1/4*x^2*y - 1/4*y^3 - 1/4*y*z^2 - 3/4*y,
    d*z^2 + 3*d + 1/4*x^2*y + 1/4*y^3 + 1/4*y*z^2 - 13/4*y,
    x^4 + 2*x^2*y^2 + 2*x^2*z^2 - 10*x^2 + y^4 + 2*y^2*z^2
    - 10*y^2 + z^4 + 6*z^2 + 9]

Thus the corresponding variety is defined by the polynomial \[x^{4} + 2 x^{2} y^{2} + 2 x^{2} z^{2} - 10 x^{2} + y^{4} + 2 y^{2} z^{2} - 10 y^{2} + z^{4} + 6 z^{2} + 9.\]

Let \(I\) be the ideal defined by the given equations. We compute the remainder of each expression modulo a Gröbner basis for \(I\). Using Sage, we find:

sage: R.<a,b,c> = PolynomialRing(QQ)
sage: I = ideal(a+b+c-3, a^2+b^2+c^2-5, a^3+b^3+c^3-7)
sage: I.reduce(a^4+b^4+c^4)
9
sage: I.reduce(a^5+b^5+c^5)
29/3
sage: I.reduce(a^6+b^6+c^6)
19/3

Thus \(a^4+b^4+c^4-9 \in I\), which means this expression is zero whenever the values of \(a,b,c\) vanish for the equations above. Thus \(a^4+b^4+c^4=9\), \(a^5+b^5+c^5=\frac{29}{3}\) and \(a^6+b^6+c^6=\frac{19}{3}\).

Using Sage, we compute the appropriate elimination ideals.

sage: R.<y,x> = PolynomialRing(QQ,order="lex")
sage: I = ideal([x^2+2*y^2-3,x^2+x*y+y^2-3])
sage: I.groebner_basis()
[y + 1/2*x^3 - 3/2*x, x^4 - 4*x^2 + 3]
sage: R.<x,y> = PolynomialRing(QQ,order="lex")
sage: I = ideal([x^2+2*y^2-3,x^2+x*y+y^2-3])
sage: I.groebner_basis()
[x^2 + 2*y^2 - 3, x*y - y^2, y^3 - y]

Thus \(I \cap k[x] = \langle x^4 - 4x^2+3 \rangle\) and \(I \cap k[y] = \langle y^3-y \rangle\).

We use Sage to find a lex Gröbner basis:

sage: R.<x,y,z> = PolynomialRing(QQ, order="lex")
sage: ideal([x^2+y^2+z^2-4, x^2+2*y^2-5, x*z-1]).groebner_basis()
[x + 2*z^3 - 3*z, y^2 - z^2 - 1, z^4 - 3/2*z^2 + 1/2]

From this we see that:

• \(I_1 = I \cap k[y,z] = \langle y^{2} - z^{2} - 1, z^{4} - \frac{3}{2} z^{2} + \frac{1}{2} \rangle\)
• \(I_2 = I \cap k[z] = \langle z^{4} - \frac{3}{2} z^{2} + \frac{1}{2} \rangle\)

By inspection (or via the quadratic formula) we find the roots of \(z^{4} - \frac{3}{2} z^{2} + \frac{1}{2}\) are \(\pm 1\) and \(\pm \frac{\sqrt{2}}{2}\). We are interested in only the rational roots so we have \(z = \pm 1\). If \(z=\pm 1\), then \(y^{2} - z^{2} - 1=y^2-2\) has no rational roots. Thus the system has no rational solutions.