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Four problems

Ideal description problem

[bluebox=Ideal description problem] Does every ideal $I\subseteq k[x_1, \ldots, x_n]$ have a finite generating set? That is, can we write $I = (f_1, \ldots, f_s)$ for $f_i \in k[x_1, \ldots, x_n]$?
[/bluebox]

Ideal membership problem

[bluebox=Ideal membership problem] Given $f \in k[x_1, \ldots, x_n]$ and an ideal $I = (f_1, \ldots, f_s)$, determine if $f \in I$. Notice that this is related to determining if the variety $V(f_1, \ldots, f_s)$ lies on the variety $V(f)$. [/bluebox]

Problem of solving polynomial equations

[bluebox=Problem of solving polynomial equations] Find all common solutions in $k^n$ of a system of polynomiale equations (i.e. the set of points of an affine variety). [/bluebox]

The implicitization problem

[bluebox=The implicitization problem] Let $V \subseteq k^n$ be given parametrically as

for $i = 1, \ldots, n$. If the $g_i$ are polynomials (or rational functions) in the variables $t_j$, then $V$ will be an affine variety or part of one. Find a system of polynomial equations (in the $x_i$) that defines the variety. [/bluebox]

Example 1 ($n = 1$)

When $n = 1$, the ideal description problem is trivial as $k[x]$ is a Euclidean domain and, therefore, a PID.

The membership problem is also pretty easy. Given that $I = (g)$, we just divide $f$ by $g$ and check for if there is a remainder or not.

Example 2 ($n$ arbitrary, linear system)

Now let $n$ be arbitrary. From linear algebra, we know that it is easy to solve a linear $m\times n$ system:

where $A = [a_{ij}] \in k^{m\times n}$ and $b\in k^m$. In particular, we just need to RREF $Ax - b = 0$.

Example 3 ($n$ arbitrary, linear system)

Let $V \subseteq k^n$ be parametrized by

where $A [a_{ij}] \in k^{m\times n}$, $b \in k^m$, $x\in k^m$, and $t \in k^n$ is the parameter. Our goal is to consider the implicitization problem for a simple system.

[example] Let $V \subseteq k^4$ be the affine linear subspace parametrized by

Thinking of the system as a system in $t_1, t_2$ and $x_1 \ldots x_4$, we subtract the $x_i$ terms and constant terms to obtain the augmented matrix

Then compute an RREF to get

The last two rows define $V$ in $k^4$. [/example]