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Let $f(x) \in \mathbb{Z}[x]$ be a polynomial. I would like to have an estimate for the number of representations $R(n)$ of $n \in \mathbb{Z}$ in the form $$ f(x) - f(y) = n, \qquad x,y \in \mathbb{N}. $$ If $f(x)=x^2$, then the number of representations is a classical problem, and $R(n) = \mathcal{O} (|n|^\varepsilon)$. Is the same true when $f(x)$ is an other polynomial of degree 2 or higher?

(If you have an answer, please include a reference or good argument. Thanks.)

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migrated from mathoverflow.net Sep 24 '15 at 21:52

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    $\begingroup$ I assume you mean $f(x)-g(y)$, for otherwise $x-y$ is a factor. See mathoverflow.net/a/14084/806 for reducibility of $f(x)-g(y)$ over $\mathbb{C}$. $\endgroup$ – Boris Bukh Sep 24 '15 at 15:07
  • $\begingroup$ The comment above answers to a previous (oversimplified) form of the question, which was really stupid. I have changed the question - now it is what I really want to know. $\endgroup$ – Kurisuto Asutora Sep 24 '15 at 15:31
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    $\begingroup$ The modified version is also not hard. Factor $f(x+d)-f(x)$ as $d\cdot (f(x+d)-f(x))/d$. The number $d$ must be a factor of $n$ (only $O(n^{\varepsilon})$ choices), and once $d$ is fixed the other factor must be equal to $n/d$. As the other factor is a polynomial of positive degree in $x$, the result follows. $\endgroup$ – Boris Bukh Sep 24 '15 at 15:37

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