I am trying to find polynomials $f \in \mathbb{F}_{q} [x_1, x_2, \dots, x_m]/(x_i^p-x_i)$ such that $f-1=0$ has precisely a given number of roots.
For example, $f(x, y)$ in $\mathbb{F}_5[x, y]/((x_1^5-x_1)(x_2^5-x_2))$ with exactly $2$ roots.
Is it always possible? Are there any upper and/or lower bounds that can tell when it is impossible?
P.S. I only know of the bound by Schwartz-Zippel Lemma.
Here are some more details. In fact we can prove the stronger statement that every function $\mathbb{F}_q^m \to \mathbb{F}_q$ is represented by a polynomial (which is even unique if we ask for the degree in each variable to be less than $q$). By linearity it suffices to show that a function which is zero everywhere except at a specific point is a polynomial, and by translating it suffices to show that a function which is zero everywhere except at the origin is a polynomial. This polynomial is easy to construct: it is just
$$(1 - x_1^{q-1})(1 - x_2^{q-1}) \dots (1 - x_m^{q-1}).$$
By Fermat's little theorem (or rather the mild generalization to finite fields, I don't know if it has a name) this polynomial vanishes unless $x_1 = x_2 = \dots = x_m = 0$, and at the origin it is equal to $1$.
