Counting integral points on a surface

Question

Let $f$ be a homogeneous polynomial with integral coefficients of 4 variables $a$, $b$, $c$ and $d$. Suppose $f$ is invariant under the rotation that rotates $(a,b)\in\mathbb{R}^2$ and $(c,d)\in\mathbb{R}^2$ simultaneously by the same angle (so this is a diagonal $SO(2)$ action on $\mathbb{R}^2\times\mathbb{R}^2$.) Let $V=f^{-1}(0)$. Let $R>0$, and $B_R$ be the ball of radius $R$ in $\mathbb{R}^4$. What is the smallest exponent $n$ one can have in the following inequality:

$$ \#(V\cap B_R\cap \mathbb{Z}^4)\le C_{f,\epsilon}R^{n+\epsilon},$$

where $C_f$ is a constant that only depends on $f$ and $\epsilon$? In particular, can we take $n=2$? My guess comes from the few examples that I try: $ac+bd$, $ad-bc$ and linear combinations of $a^2+b^2$ and $c^2+d^2$. In these cases we can use the divisor bound $\tau(n)\le C_\epsilon n^\epsilon$, but in general there doesn't seem to be the sort of factorization that makes the divisor bound applicable.

Answer 1

This question was answered by Heath-Brown in his paper "The density of rational points on curves and surfaces", Annals of Mathematics 155 (2002), 553-598. In particular, Theorem 9 in this paper asserts the following. Let $F \in \overline{\mathbb{Q}}[x_1, x_2, x_3, x_4]$ be a homogenenous polynomial which is irreducible over $\overline{\mathbb{Q}}$. Put $N(F;B)$ for the cardinality of the set $\{(x_1, x_2, x_3, x_4) \in \mathbb{Z}^4 : F(x_1, x_2, x_3, x_4) = 0, \max_{1 \leq i \leq 4} |x_i| \leq B\}$. Then for any $\epsilon > 0$

$$\displaystyle N(F ;B) = O_\epsilon(B^{2 + \epsilon}).$$

Moreover, the implied constant, as suggested in the notation, is dependent only on $\epsilon$ and is independent of $F$.

One can also show that when $d = \deg F \geq 3$, that the bulk of the integer points lie on lines contained on the surface $X$ defined by $F = 0$. For example, when $d = 3$ and if $N_1(F;B)$ for the number of integer points counted by $N(F;B)$ which do not lie on any line contained in $X$, then

$$\displaystyle N_1(F;B) = O_\epsilon(B^{12/7 + \epsilon}).$$

Finally, when the number of variables $n > 4$, the result

$$\displaystyle N(F;B) = O_{d, n,\epsilon} \left(B^{n-2+\epsilon}\right)$$

was established by Salberger when $d \geq 4$. Salberger's proof shows that the implied constant is independent of $F$ and depends only on $n,d,\epsilon$. Salberger also established the exponent $n-2$ for $d = 3$, but the proof in that case is not uniform in $F$.