Do polynomial values rarely have large multiple prime factors?

Question

I am interested in the following set-up: Let $F \in \mathbb{Z}[x_1,\dots,x_n]$ be a fixed irreducible homogeneous polynomial of degree $d$ and consider the quantity $$N_{\delta}(B)=\#\{(x_1,\dots,x_n) \in \mathbb{Z}^n: \vert x_1\vert ,\dots,\vert x_n\vert \le B, \exists p \ge B^{\delta}: p^2 \mid F(x_1,\dots,x_n)\}$$ where $\delta$ is some (small) positive constant. Of course the trivial bound is $N_{\delta}(B) \ll B^n$ and my question is whether we can get a power saving $$N_{\delta}(B) \ll B^{n-c(\delta)}$$ as $B \to \infty$, for some positive constant $c(\delta)$, possibly depending on $F,n,d$ as well. Heuristically, the probability that a number is divisible by $p^2$ is $\frac{1}{p^2}$, so we would expect $\frac{B^n}{p^2}$ values divisible by $p^2$ so that we could hope for a bound of the type $$N_{\delta}(B) \ll \sum_{p \ge B^{\delta}} \frac{B^n}{p^2} \ll B^{n-\delta}.$$ But of course this is very far from being a rigorous argument...

(It definitely seems like a question that has to have appeared before, but unfortunately I could not locate any reference.)

Addendum (31.05.): If this seems to hard to answer, does anyone at least have an intuition

a) whether or not this ought to be true (or whether there is something obvious I am missing) and

b) if true, whether or not this ought to be in reach of existing technology?

Answer 1

The question of showing that a multivariable polynomial takes on many squarefree values, or more generally controlling the size of the largest square factor, is referred to in the literature as the "squarefree sieve".

As you have already observed, standard sieve methods only allow you to filter out prime factors up to a certain size -- so one needs an additional ingredient to get control over the case of large prime factors. Generally this additional ingredient has come from diophantine geometry. There's various work out there, starting with that of Granville in the one-variable case, obtaining an exact asymptotic for the squarefree sieve conditional on the abc conjecture. Without the abc conjecture, things get harder, and unconditional results are mainly known in low degree cases or where the polynomial has additional symmetries that can be exploited.