Simplest Case: 1 variable

Find solutions to

$$f(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0 = 0.$$

with $x\in\mathbf{Q}$. (Here $a_0,a_1,\ldots\in\mathbf{Z}$.)

Rational root theorem: If $x=\frac{b}{c}$ is a solution with $b,c\in\mathbf{Z}$ having no common factor, then $c\mid a_n$ and $b\mid a_0$. So it is easy to determine all solutions in $\mathbf{Q}$ to a polynomial in $1$ variable.

(Proof:

$$a_n b^n + a_{n-1}b^{n-1}c + \cdots + a_1 b c^{n-1} + a_0 c^n = 0$$

implies that $c\mid a_nb^n$ so $c\mid a_n$ and $b\mid a_0 c^n$ so $b\mid a_0$.)

Also, the number of solutions to $f(x)=0$ is at most $n$.