upper bound for the ratio of two solutions of a Diophantine equation

Question

Let $P(x)$ be a polynomial of degree $d$ with integer coefficients. Define $$ A=\{(m,n)\in (\mathbb{Z}\setminus\{0\})^2: m-n=P(m)-P(n) \} $$ and $$ L=\max_{(m,n)\in A}\left|\frac{m}{n}\right| $$ I can prove that $L$ is bounded by some constant depending on $P$. But I'm wondering if $L$ is bounded by some constant depending only on $d$, the degree of $P$.

Answer 1

The answer is no. For example, let $d=2$. Consider the quadratic polynomial $P(x)=(x-1)(x-n)$. Then $P(m)=0$ has two solutions, who ratio goes to infinity as $n$ goes to infinity.