2
$\begingroup$

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......

| cite | improve this question | | | | |
$\endgroup$
4
$\begingroup$

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.

| cite | improve this answer | | | | |
$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – var Mar 3 at 9:23
  • $\begingroup$ 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? $\endgroup$ – var Mar 3 at 9:55

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.