Rewrite the polynomial \[x^5y^5z + 3x^5y^4z^2 + 5x^6y^4 + 7x^4y^6z\] using each of the lex, grlex and grevlex orders.

Without using computer algebra, compute \[\gcd(x^4 + 3x^3 + 2x^2 + x - 1, x^4 + x^2 + 1).\]

Consider the following polynomials: \[\begin{align*} f &= x^3y - 2x^2y^2 + xy^3\\ g_1 &= x^2y - xy\\ g_2 &= xy^2 - y^3\\ g_3 &= y^2 - 2y \end{align*}\] Without using computer algebra, compute the remainder of \(f\) on division by the ordered set \(G=(g_1,\ g_2,\ g_3)\) using the division algorithm and the lex order.

Consider the variety \(V \subset \mathbb{C}^3\) given by the parametrization \[\begin{align*} x &= t^2-t\\ y &= t^3\\ z &= t^2\ . \end{align*}\] where \(t\) is a parameter. Find generators for an ideal \(I\) such that \(V=\mathbf{V}(I)\).

Consider the ideal \(I \subseteq \mathbb{C}[x,y,z]\) where \[I = \langle x^2y^2 - x^2z^2 - yz,\ y^2 + z^2 - 1 \rangle.\] Describe the set of all partial solutions \((y,z)=(a_2,a_3)\) that extend to full solutions \((x,y,z)=(a_1,a_2,a_3) \in \mathbf{V}(I)\).

Consider the ideals \(I = \langle x,y^2 \rangle\) and \(J = \langle x-2y \rangle\) in \(\mathbb{Q}[x,y]\). Find \(\sqrt{I}\), \(I+J\), \(I\cdot J\), \(I \cap J\), \(I:J\) and \(I:J^\infty\).

Suppose that \(I,J\) are ideals of \(k[x_1,\ldots,x_n]\) such that \(1 \in I+J\). Prove that \(IJ = I \cap J\).

Let \(I,J,K\) be ideals of \(k[x_1,\ldots,x_n]\). Prove that \((I:J):K=I:JK\).

Find an example of each of the following:

• Ideals \(I,J \subseteq \mathbb{C}[x,y]\) such that \(I \ne J\), but \(\mathbf{V}(I)=\mathbf{V}(J)\).
• An ideal \(I \subseteq \mathbb{R}[x,y]\) such that \(\mathbf{V}(I) = \emptyset\), but \(I \ne \mathbb{R}[x,y]\).
• Radical ideals \(I,J \subseteq \mathbb{C}[x,y]\) such that \(I+J\) is not radical.
• Ideals \(I,J \subseteq \mathbb{C}[x,y]\) such that \(IJ \ne I \cap J\).