Math and stuff

Proposition Let $I$ and $J$ be ideals in $k[x_1, \cdots, x_n]$, where $k$ is an arbitrary field. Prove the following: If $I^l \subset J$ for some integer $l > 0$, then $\sqrt{I} \subset \sqrt{J}$. $\sqrt{I + J} = \sqrt{\sqrt{I} + \sqrt{J}}$. Solution 1 Let $f \in \sqrt{I}$. Then $f^k \in...

Proposition Let $I$ be an ideal in $k[x_1, \cdots, x_n]$. In the special case when $\sqrt{I} = \ev{f_1, f_2}$, with $f_i^{m_i} \in I$, prove that $f^{m_1 + m_2 - 1} \in I$ for all $f \in \sqrt{I}$. Now prove that for any $I$, there exists a single integer $m$ such...

Proposition Find bases for the elimination ideals $I_1$ and $I_2$ for the ideal $I$ determined by the equations: \[\begin{align*} x^2 + y^2 + z^2 &= 4, \\ x^2 + 2y^2 &= 5, \\ xz &= 1. \end{align*}\] How many rational solutions are there? Solution from sympy import * from sympy.polys.orderings...

Proposition Let $V = V(y - x)$ in $\mathbb{R}^2$ and let $\phi: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ be the polynomial mapping represented by $\phi(x, y) = (x^2 - y, y^2, x - 3y^2)$. The image of $V$ under $\phi$ is a variety in $\mathbb{R}^3$. Find a system of equations defining the image...

Proposition Let $f_1 = y^2 + 2xy - 1$ and $f_2 = x^2 + 1$. Prove that $\ev{f_1, f_2}$ is not a radical ideal. Solution $f_1 + f_2 = (x + y)^2$. Thus $(x + y)^2 \in \ev{f_1, f_2}$. We claim that $x + y \notin \ev{f_1, f_2}$. from sympy...