In this paper of Erdos and Mahler, they claim the following result and say that it will be published in future work but I couldn’t find the future work. In fact, I only need a proof of an application of this in their paper, namely lemma 8 in the link, but the application hinges heavily on this unproved claim. So here’s the claim (Lemma 7 in the link)
Let $F$ be a degree $n$ binary form with integer coefficients and nonzero discriminant and $t$ is a positive integer. Then, consider the pairs $(x,y)$ for which
- $F(x,y)$ does not vanish, and
- For primes $p_1,\cdots p_t$ such that the largest possible product of the form $p_1^{e_1}\cdots p_t^{e_t}$, divides $ F(x,y)$, denote this product by $Q(x,y)$. Suppose that the pair satisfies $$ |\frac{ F(x,y) }{Q(x,y)}|< |x,y|^{\frac{1}{2}n-1-\frac{1}{28}}.$$
Then, the number of such pairs is bounded by $C^{1+t}$ for some constant only depending on $F$.
Id like to find this reference or give an argument for why this is true? Or even if I can find a reference for lemma 8 (that hopefully isn’t too involved) because I need to reproduce this proof in another setting.