Math and stuff

Proposition Let $I \subset k[x_1, \cdots, x_n]$ be an ideal, and let $f_1, \cdots, f_s \in k[x_1, \cdots, x_n]$. Prove that the following statements are equivalent: $f_1, \cdots, f_s \in I$. $\ev{ f_1, \cdots, f_s } \subset I$. Solution Since $I$ is closed under addition and closed under multiplication by...

Proposition Given $f \in k[x]$, find a parametrization of $V(y - f(x))$. Solution $x = t, y = f(t)$.

Proposition Sketch the following affine varieties in $\mathbb{R}^2$: $V(x^2 + 4y^2 + 2x - 16y + 1)$. $V(x^2 - y^2)$. $V(2x + y - 1, 3x - y + 2)$. Solution 1 $x^2 + 4y^2 + 2x - 16y + 1 = 0 \implies (x + 1)^2 + 4(y -...

Proposition Let $I = \ev{ g_1, g_2, g_3 } \subset \mathbb{R}[x, y, z]$, where $g_1 = xy^2 - xz + y, g_2 = xy - z^2$ and $g_3 = x - yz^4$. Using the lex order, give an example of $g \in I$ such that $\LT(g) \notin \ev{ \LT(g_1), \LT(g_2),...

Proposition Let $I \subset k[x_1, \cdots, x_n]$ be an ideal with the property that for every $f = \sum c_{\alpha} x^{\alpha} \in I$, every monomial $x^{\alpha}$ appearing in $f$ is also in $I$. Show that $I$ is a monomial ideal. Solution Let $A = \{ \alpha \mid \sum c_{\alpha} x^{\alpha}...