Let $X\hookrightarrow\mathbb P^n_{\mathbb F_q}$ be a geometrically integral hypersurface over the finite field $\mathbb F_q$ of degree $\delta$. In order to estimate the number of its rational points, we have $$|\#X(\mathbb F_q)-(q^{n-1}+\cdots+1)|\leq(\delta-1)(\delta-2)q^{n-\frac{3}{2}}+C(n,\delta)q^{n-2}.$$ My question is about the order of $\delta$ in the constant $C(n,\delta)$. In fact, we have $C(n,\delta)\ll_n\delta^4$ uniformly, and when $q$ is large enough, we can prove $C(n,\delta)\ll_n\delta^2$. Can we prove $C(n,\delta)\ll_n\delta^2$ uniformly for all $q$?

If it is too difficult, we can consider whether we can prove $$|\#X(\mathbb F_q)-(q^{n-1}+\cdots+1)|\leq C'(n,\delta)q^{n-\frac{3}{2}},$$ where we require $C'(n,\delta)\ll_n\delta^2$ uniformly. Of course, this is OK for the case of $n=2$, but for arbitrary dimension, I don't know......


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