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Related to this question Coefficients of $f(t)=(\sum_{m=0}^{+\infty}e^{2\pi im^4t})(\sum_{m=0}^{+\infty}e^{2 \pi inm^4t})$ I have the following question:

What is the set of homogeneous polynomials $p(x,y) \in \mathbb{Z}[x,y]$ wich induces injective function when viewed as polynomial functions $p:{\mathbb{N}}^2 \to \mathbb{Z}$?

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    $\begingroup$ As with your preceding question, probabilistic heuristics (see e.g. terrytao.wordpress.com/2012/09/18/… ) suggest such polynomials should be generically injective in degree 5 and higher but not in degree 4 or lower, at least for "typical" choices of polynomial $p$. It may be possible to use the Bombieri-Lang conjecture to make the notion of "typical" precise (probably has to do with the the non-trivial component of the variety $p(x,y)=p(z,w)$ being of general type). $\endgroup$
    – Terry Tao
    Commented Feb 24, 2015 at 21:17
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    $\begingroup$ Somewhat related: it is an unsolved problem of Erdos to find a polynomial $p$ such that all the sums $p(a)+p(b)$, $0\le a<b$, are distinct. This is F30 in Guy, Unsolved Problems In Number Theory. $\endgroup$
    – Gerry Myerson
    Commented Feb 24, 2015 at 22:30
  • $\begingroup$ @TerryTao: Thanks for the comment and for the reference. Gerry Myerson: Interesting! Thanks for the reference $\endgroup$
    – Marcel1994
    Commented Feb 25, 2015 at 0:47

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