I am wondering if there is any research on the lower bound of $\mathbb{F}_q$-points of a smooth hypersurface with degree $d$ in projective space $\mathbb{p}^n_{\mathbb{F}_q}$?
To be precise, let $X$ be a degree $d$ smooth hypersurface in $\mathbb{P}^n_{\mathbb{F}_q}$ (here we can assume that $X$ is given by a homogeneous equation with $\mathbb{F}_q$ coefficients). Denote by $X(\mathbb{F}_q)$ the set of points of $X$ which are invariant under the natural $\mathbb{F}_q$-Frobenius action. I am wondering if there is any lower bound of $\#X(\mathbb{F}_q)$ in terms of $n$ and $d$?
Of course there is always a trivial bound, say $\#X(\mathbb{F}_q)\geq 0$. Sometimes, the Hasse-Weil bound provide some information. But I am looking for a more elementary bound such as the one in Homma-Kim's "An elementary bound for the number of points of a hypersurface over a finite field".
Any suggestion/comment is appreciated.