  Let $X^n$ be a nonsingular hypersurface of degree $d\geq 2$ 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 $n\geq 5$ and
$d\leq q$.
