Hi,

After the response to the following question, Rational roots to quadratic forms in 4 variables, I am now considering the following question.

Let $d \geq 2$ be a positive integer, and suppose that $F(x_1, x_2, x_3, x_4)$ is a form of degree $d$ with integer coefficients. Let $N(F,B)$ be the number of primitive integer solutions $(x_1, x_2, x_3, x_4)$ to $F(x_1, x_2, x_3, x_4) = 0$ (where primitive means that the $\gcd$ of all non-zero entries is 1, that at least one of the entries is non-zero, and that the smallest index $i$ such that $x_i \ne 0$ satisfies $x_i > 0$) satisfying $|x_i| \leq B$, for some $B > 0$. Let $\lVert F \rVert$ be the maximum of the absolute values of the coefficients of $F$.

I am wondering if a theorem of the following variety exists: In the above situation, we either have $N(F,B) \ll_\epsilon B^{\theta + \epsilon}$ for some small $\theta$, or we have $\lVert F \rVert \ll_{d, \epsilon} B^{u(d, \epsilon)}$ where $u$ is some positive function of $d$ and $\epsilon$.

As a comparison, in the three variable case we have the following result due to Bombieri and Pila, in E. Bombieri and J. Pila, "The number of integral points on arcs and ovals", Duke Mathematical Journal., 59 (1989), 337-357 which states that if $F(x,y,z)$ is a form of degree $d$, then either $N(F,B) \ll d^2$ or $\lVert F \rVert \ll B^{d(d+1)(d+2)/2}$.

The heuristic here is that if $F$ has lots of solutions in the box $|x_i| \leq B$, then its coefficients cannot be too big because then even slight variations to the coordinates would throw the value of $F$ outside the box.

Any help would be greatly appreciated.