Strengthening an implication of the abc conjecture

Question

Granville gives p.5 an implication of the abc conjecture:

Assume the abc conjecture. Let $f(x,y)$ be squarefree homogeneous polynomial with integer coefficients. For coprime integers $m,n$ if $q^2 \mid f(m,n)$ then $q \ll \max(|m|,|n|)^{2+\epsilon}$.

Can we strengthen this to $q \ll |mn|^{1+\epsilon}$?

For constant $n$ this is consistent with the paper.

Are there heuristic arguments for small roots modulo squares?

Answer 1

Yes. Without loss of generality, $x$ and $y$ divide $f(x,y)$. (If not, then multiply by one or the other, and $q$ will still divide it).

Without loss of generality $m \geq n$. Then we know that the product of primes dividing $f(m,n)$ is at least $m^{\deg f - 2 - \epsilon}$ and that $f(m,n)$ is at most a constant times $m^{\deg f -1} n$, so $q$ is at most the ratio, which is $ (mn)^{1+\epsilon}$.