Let $F(x_1, \cdots, x_n) \in \mathbb{Z}[x_1, \cdots, x_n]$ be a homogeneous polynomial, irreducible over $\mathbb{Q}$. Let $A$ be a positive constant, and let $B$ be a positive real number understood to be large (it is a parameter tending to infinity). Let $l$ be a square-free number whose prime divisors are all less than $\Xi = B^{1/(\log \log B)^2}$ and that $l \leq B^{1/6}$. Let $\omega^\dagger(m)$ denote the number of prime divisors of $m$ which exceed $\Xi$. If $n = 2$, then it is shown as Lemma 10.2 in Hooley's paper "On the power-free values of polynomials in two variables" in Roth's 80th birthday volume that for any $Y$ satisfying $B^{1/2} < Y \leq B$, that there exist positive numbers $C_1, C_2$ such that

$$\displaystyle \sum_{\substack{(u_1, u_2) \equiv (a_1, a_2) \pmod{l} \\ |u_1|,|u_2| \leq Y}} A^{\omega^\dagger(F(u_1, u_2))} \leq C_1 Y^2 (\log \log B)^{C_2} l^{-2}.$$

Hooley claims that this follows from van der Corput's method, but the paper he cited is in a language I cannot read, and otherwise I am not familiar with van der Corput's method to think about the higher dimensional analogue of this. Does a similar statement hold for $n > 2$?


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