Does multivariate polynomial over a finite field always have a solution (in the field)? Let $K = F_{p^e}$ be a finite field. Say I have a single polynomial $f \in K[x_1,\ldots, x_n]$ of degree $d$.
Under what conditions on $n$ and $d$ can I claim that a root to $f$ always exists? In other words, do there exist polynomials over finite fields with an arbitrary number of variables and bounded degree that have no roots in the field?
If a root exists, can I compute it efficiently?
The polynomial
$$
f=(x_1+x_2+\cdots +x_n)^2−k
$$
where $k$ is not a quadratic residue modulo $q$ has no roots in  the finite field $\Bbb F_{q}$.
Even with $n=1$, this would mean the field is algebraically closed, and an algebraically closed field cannot be finite.