Write, for $f(x) = x^d + a_2 x^{d-2} + \cdots + a_d\in \mathbb{Z}[x]$, $H(f) := \max(|a_i|^{\frac{1}{i}})$.

Does anyone know of a reference that would allow me to show that the proportion of $f$ with $\deg{f} = d > 4, H(f)\leq X$, and discriminant having a square factor $> X^\delta$ is $O(X^{-c})$ for some $c = c(d,\delta) > 0$?

The proportion of polynomials in $\mathbb{Z}_p[x]$ with discriminant divisible by $p^2$ is $O(p^{-2})$, so that handles primes $p\leq X$. There has been so much work on squarefree discriminants of polynomials that this must be straightforward given the methods, but unfortunately I don't know them.

Thanks much!!

**Edit**:

I found a paper of Bhargava on the "geometric sieve", and I can almost answer the question by using his methods, except his condition 6 (and subsequent trick to reduce to a higher-codimension variety) doesn't apply. He solves this problem for $d=4$ via something that seems special to that case (he calls it the "embedding sieve" --- it's at the end of the paper). Does anybody know if there is a similar embedding available for higher $d$? I suppose this is now all of a sudden a question in invariant theory, which I essentially know nothing about. Thanks again!