I am interested in the ternary case of a theorem of 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. The following statement is found in D.R. Heath-Brown's work, namely D.R. Heath-Brown, "Counting rational points on algebraic varieties", Analytic number theory, 51-95, Lecture Notes in Mathematics., 1891, Springer, Berlin, 2006.
Theorem 5: Suppose $F(x_1, x_2, x_3)$ is a non-zero form of degree $d$ with integer coefficients and content 1. Define $N(F,B)$ to be the number of primitive triplets $(x_1, x_2, x_3)$ (primitive means $\gcd(x_1, x_2, x_3) = 1$) such that $F(x_1, x_2, x_3) = 0$, and that $|x_i| \leq B$ for $i = 1, 2, 3$. Also define $\lVert F \rVert$ to be the maximum modulus of the coefficients of $F$. Then either $N(F,B) \ll d^2$ or $\lVert F \rVert \ll B^{d(d+1)(d+2)/2}$.
I am interested in an analogous result for forms with four variables. Namely, a bound in terms of the degree of the form for $N(F,B)$, and alternatively a bound in terms of $B$ and $d$ of $\lVert F \rVert$. In the same Heath-Brown paper he gave several results which bounds $N(F,B)$ with a bound in terms of both $B$ and $d$, which (a priori) is not what I am looking for. Theorem 14 and 15 in the Heath-Brown paper shows that one can bound $N(F,B)$ by counting the number of zeroes of $k$ polynomials, but the key is then controlling the number of zeroes of those polynomials. Any insights would be greatly appreciated.
