Math and stuff

Proposition Prove that $f \in k[x, y, z]$ is symmetric if and only if $f(x, y, z) = f(y, x, z) = f(y, z, x)$. Solution Consider $S \subset S_3$ be the set of all permutations under which $f$ is invariant. It suffices to show $S = S_3$. Since $S$...

Proposition Let $C$ be the twisted cubic curve in $k^3$. Show that $C$ is a subvariety of the surface $S = V(xz - y^2)$. Find an ideal $J \subset k[S]$ such that $C = V_S(J)$. Solution 1 Let $f(x, y, z) = xz - y^2$. $C = \{ (a, a^2,...

Proposition Show that $\phi_1(x, y, z) = (2x^2 + y^2, z^2 - y^3 + 3xz)$ and $\phi_2(x, y, z) = (2y + xz, 3y^2)$ represent the same polynomial mapping from the twisted cubic in $\mathbb{R}^3$ to $\mathbb{R}^2$. Solution $\{ (a, a^2, a^3) \mid a \in \mathbb{R} \} \subset \mathbb{R}^3$ is...

Proposition Prove the Geometric Extension Theorem using the Extension Theorem and Lemma 1. (Ideals, Varieties, and Algorithms) Solution First, we will show that $\pi_1(V) \cup (V(c_1, \cdots, c_s) \cap V(I_1)) \subset V(I_1)$. $\pi_1(V) \subset V(I_1)$. Lemma 1 on P.129 (Ideals, Varieties, and Algorithms) $(V(c_1, \cdots, c_s) \cap V(I_1)) \subset V(I_1)$....

Proposition Show that $\mathbb{R}[x]/\ev{x^2 - 4x + 3}$ is not an integral domain. Solution $x - 3 + \ev{x^2 - 4x + 3}, x - 1 + \ev{x^2 - 4x + 3}$ are both nonzero elements. However, their product is $0$.