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Suppose that $F(x_1, \cdots, x_n) \in \mathbb{Z}[x_1, \cdots, x_n]$ is a polynomial of degree $d$, and examine the quantity $$\displaystyle N(F;X, B) = \# \{(x_1, \cdots, x_n) \in \mathbb{Z}^n | -X \leq F(x_1, \cdots, x_n) \leq X, B < \max(|x_1|, \cdots, |x_n|) \leq 2B\}. $$

In the $n=2$ case, looking at the curve $x - y^d = 0$ we see that $N(F; X, B) \gg_B X^{1/d}$. This continues to be the case for binary forms, as was shown by Erdős and Mahler in 1938. Heath-Brown obtained the result that most of the representations are essentially unique.

It seems intuitive that $N(F;X,B)$ should be of order of magntitude $X^{n/d}$ for any dimension, but I was not able to locate any references. Any help would be appreciated.

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