Math and stuff

Proposition When $k = \mathbb{C}$, the conclusion of Theorem 1 (P.134, Ideals, Varieties and Algorithms) can be strengthened. Namely, one can show that there is a variety $W \subsetneq V(I_m)$ such that $V(I_m) \setminus W \subset F(\mathbb{C}^m)$. Prove this using the Closure Theorem. (P.131, Ideals, Varieties and Algorithms). Solution Let...

Proposition Give an example to show that the result is false over $\mathbb{R}$. Solution Let $F: \mathbb{R} \rightarrow \mathbb{R}$ be defined such that $F(t) = t^2$. Let $I = \ev{x - f_1(t)} = \ev{x - t^2}$. Let $V = V(I) = \{ (a, a^2) \mid a \in \mathbb{R} \}$. Then...

Proposition Suppose that $k$ is a field and $\phi: k[x_1, \cdots, x_n] \rightarrow k[x_1]$ is a ring homomorphism that is the identity on $k$ and maps $x_1$ to $x_1$. Given an ideal $I \subset k[x_1, \cdots, x_n]$, prove that $\phi(I) \subset k[x_1]$ is an ideal. Solution Since $\phi$ is a...

Proposition Prove that the diagram (3) on P.134 (Ideals, Varieties, and Algorithms) commutes and $i(k^m) = V$. Solution For any $(t_1, \cdots, t_m) \in k^m$, \[\begin{align*} (\pi_m \circ i)(t_1, \cdots, t_m) &= \pi_m(t_1, \cdots, t_m, f_1(t_1, \cdots, t_m), \cdots, f_n(t_1, \cdots, t_m)) \\ &= (f_1(t_1, \cdots, t_m), \cdots, f_n(t_1, \cdots,...

Proposition Verify that $\ev{(y - z)x^2 + xy - 1, (y - z)x^2 + xz - 1} = \ev{xy - 1, xz - 1}$. Also check that $y - z$ vanishes at all partial solutions in $V(I_1)$. Solution $\ev{(y - z)x^2 + xy - 1, (y - z)x^2 + xz...