An elementary bound for the number of points of a hypersurface over a finite field
Given a plane curve \(C\) of degree \(d\) without linear components defined over a finite field \(\mathbb F_q\), P. Sziklai conjectured that the number of rational points of \(C\) is bounded by \[ N_q(C)\leq (d-1)q+1. \] \textit{M. Homma} and \textit{S. J. Kim} [Finite Fields Appl. 15, No. 4, 468--474 (2009; Zbl 1194.14031); Contemp. Math. 518, 225--234 (2010; Zbl 1211.14037); Finite Fields Appl. 16, No. 5, 315--319 (2010; Zbl 1196.14030)] proved that this upper bound holds except for a curve of degree 4 over \(\mathbb F_4\). In this paper, the authors generalize the upper bound for a hypersurface \(X\subset \mathbb P^n\) of degree \(d\) without linear components over \(\mathbb F_q\), \[ N_q(X)\leq (d-1)q^{n-1} +dq^{n-2}+\theta_q(n-3), \] where \(\theta_q(r)=(q^{r+1}-1)/(q-1)\) is the number of points of the \(r\)-dimensional finite projective space \(\mathbb P^r(\mathbb F_q)\). Moreover, if \(n\geq 3\) they show that the equality holds at least for two irreducible hypersurfaces of different degrees. The authors also present a comparison of this bound with several well known bounds as the Weil conjecture established by P. Deligne, the Segre-Serre-Sørensen bound and its generalization given by K. Thas.
scientific article; zbMATH DE number 6146998