Equations with few solutions

Question

The only solution to the equation $x^2 + y^2$ over $\mathbb{F}_q^2$ when $q \equiv 3 \;(\bmod \;4)$ is when $x = y = 0$. Is there an analogous example in $\mathbb{F}_q^3$? That is, are there polynomial equations $f(x,y,z)$ whose only solution over $\mathbb{F}_q^3$ is when $x = y = z = 0$? For four variables, it is easy to produce an example, namely, $(x^2 + y^2)^2 + (z^2 + w^2)^2$, but I couldn't see how to proceed for three variables.

Answer 1

For $q$ a prime with $q>3$, consider $x^{q-1}+y^{q-1}+z^{q-1}=0$. As $x^{p-1}=0$ or $1$, the LHS is either $0$, $1$, $2$ or $3$, and as these are distinct modulo $q$, it's only zero when $x=y=z=0$.

Answer 2

You've given the keyword algebraic geometry. The zero set of a (finite) system of equations forms a socalled variety and the system of equations forms an ideal in the underlying polynomial ring. The finite zero sets correspond one-to-one to the socalled zero-dimensional ideals. Zero-dimensional ideals have a nice characterization.