# Density of rational points over finite fields, an estimate of Lang-Weil constant

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......

PS. When $$q\ll\delta^2$$ or $$\delta^4\ll q$$, we can prove it. But when $$\delta^2 \ll q\ll \delta^4$$, I don't know......

When $$q$$ is large, one can in fact take $$C(n,\delta)=\delta+10$$; see Corollary 6 and Corollary 9 in my paper An application of random plane slicing to counting $$\mathbb{F}_q$$-points on hypersurfaces,'' https://arxiv.org/pdf/1703.05062.pdf. Explicitly, this bound holds for $$q\geq 4\delta^{13/3}$$. If you can prove that one can take $$C(n,\delta)=O(\delta^4)$$ uniformly in $$q$$, then the proof of Corollary 6 in the above paper will give $$C(n,d)=O(d)$$ for $$q\geq d^4$$.
You can also look at the paper by A. Cafure, G. Matera, Improved explicit estimates on the number of solutions of equations over a finite field, Finite Fields and Their Applications 12 (2006) 155–185, https://arxiv.org/abs/math/0405302. They prove, among other things, that one can take $$C(n,\delta)=5\delta^{13/3}$$ uniformly. They also review the literature on the problem.
In the regime when $$q$$ is small, one can use alternative bounds for $$\#X(\mathbb{F}_q)$$; for example, $$\#X(\mathbb{F}_q)\leq \delta q^{n-1}$$ is an easy bound (see, for example, Lemma 2.1 in the paper of Cafure and Matera); then, as far as the upper bound for $$\#X(\mathbb{F}_q)$$ is concerned, one can take $$C'(n,\delta)=O(\delta^2)$$ as long as $$q=O(\delta^2)$$. But then one would need to handle the range for $$q$$ between $$d^2$$ and $$d^4$$. I would guess that there are some further bounds for $$\#X(\mathbb{F}_q)$$ that would turn handy in this range.
• Actually I have read all the related papers of Cafure and Matera. For $q\ll \delta^2$, we can just use trivial estimates, for $\delta^4\ll q$, one of their papers is OK. But for $\delta^2\ll q\ll\delta^4$......that's what I want to do but I am not able to deal with it. Thank you for your answer. – var Mar 3 at 9:23
• Or we suppose $X$ is smooth over $\mathbb F_q$, or add some conditions on the dimension of its singular locus. Could you prove it? – var Mar 3 at 9:55