# How to obtain the following “trivial” bound on the number of rational points on a hypersurface?

Let $$F: \mathbb{R}^{n} \to \mathbb{R}$$ be a smooth function. Suppose $$B$$ be a closed and bounded box. I would like to obtain for fixed $$q \in \mathbb{N}$$ $$\# \{ \mathbf{a} \in \mathbb{Z}^n : F(\frac{\mathbf{a}}{q} ) = 0 , \ \frac{\mathbf{a}}{q} \in B \} < C q^{n-1},$$ where $$C$$ is independent of $$q$$.

I think this is not possible if $$F$$ has a region where it is identically zero inside $$B$$. So my question is what assumption on $$F$$ is necessary on $$B$$ so that this bound can be achieved. Any comments would be appreciated. Thank you.

• Is $F$ a polynomial, or just any $C^\infty$-function? I am sure with the latter one can come up with an example that hits way more rational points than it should – Stanley Yao Xiao Jul 8 at 1:55
• $F$ is any $C^{\infty}$-function here. Yes, you are right, for example $F \equiv 0$. That's why I was hoping for some conditions I could put on $F$ such that this bound holds. One condition one could assume is that $F$ is non-singular on $B$. Then by implicit function theorem (and using compactness) the bound follows. Maybe this is as good as one can hope for... – Johnny T. Jul 8 at 6:55