The Hasse-Weil bound implies that for any 2-variable polynomial $P(x,y)$, there exists approximately $p$ solutions in $\mathbb{F}_p$ of $P(x,y) \equiv a \pmod p$ for sufficiently large $p$, and any integer $a$.
The Chevalley Theorem gives a sufficient condition for a homogeneous $n$-variable polynomial to have nontrivial roots in $\mathbb{F}_p$.
Are there any stronger or more general results about $n$-variable polynomials in $\mathbb{F}_p $ ?
For example, is an approximation of the numbers of solutions in $\mathbb{F}_p$ for $f(x_1 , x_2 , \cdots , x_k )=a$ known, or are there any any similar results?