On a Homma-Kim conjecture for nonsingular hypersurfaces

Let $X^n$ be a nonsingular hypersurface of degree $d\ge 2$ in the projective space ${\mathbb{P}}^{n+1}$ defined over a finite field ${\mathbb{F}}_q$ of $q$ elements. We prove a Homma-Kim conjecture on a upper bound about the number of ${\mathbb{F}}_q$-points of $X^n$ for $n=3$, and for any odd integer $n\ge 5$ and $d\le q$.