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Let $X$ be a smooth affine cubic curve in $\mathbb A^2$ defined by $f(T_1,T_2)\in\mathbb Z[T_1,T_2]$ (of course $\deg(f)=3$ by definition), and $$n(f, B)=\{(x_1,x_2)\in\mathbb Z^2| |x_1|\leq B, |x_2|\leq B, f(x_1,x_2)=0\}.$$ If we require $X$ has a smooth projective closure in $\mathbb P^2$ (Thanks for the comments of Prof. Silverman), are there any uniform upper bounds of $n(f, B)$? We allow the bound depending on $B$.

PS. "Uniform" means the upper bound doesn't depend on $f$, so Siegel's integral points theorem is insufficient.

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  • $\begingroup$ Are you asking for an asymptotic (in $B$) upper bound? Do you want upper bounds for "small" $B$? What precisely are you asking for? $\endgroup$
    – Jason Starr
    Jan 1, 2023 at 12:27
  • $\begingroup$ Yes, such bounds are available. Not sure what the state of the art is now but with the determinant method one can obtain $O_\epsilon\left(B^{1/3 + \epsilon}\right)$ with the implied constant independent of $f$. $\endgroup$
    – Stanley Yao Xiao
    Jan 1, 2023 at 12:28
  • $\begingroup$ I know what you mentioned in Salberger's papers. But I believe for smooth curves, we have better bounds. $\endgroup$
    – var
    Jan 1, 2023 at 13:02
  • $\begingroup$ I asked for a uniform estimate. Of course an asymptotic result is also preferred. $\endgroup$
    – var
    Feb 15, 2023 at 7:36

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