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Let $F \in \mathbb{Z}[x,y]$ be a polynomial of degree $2n$ such that the homogeneous degree $2n$ part of $F$, say $F_{2n}$, is positive semi-definite. How does one show that for some $\delta_n > 0$ sufficiently small (and dependent only on $n$) that there exists a positive number $c_F$ (dependent on $F$) such that the region

$$\displaystyle \left\{(x,y) \in \mathbb{R}^2 : |F(x,y)| \leq c_F (x^2 + y^2)^{n - \frac{1}{2} - \delta_n} \right\}$$

contains infinitely many integer points $(x,y) \in \mathbb{Z}^2$?

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  • $\begingroup$ What reason do you have to expect this is true? $\endgroup$
    – Alison Miller
    Commented Jan 23, 2023 at 2:28
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    $\begingroup$ It looks to me like if you take F(x, y) = x^2 + y^2 then this region is a disc of some radius (depending on c_F and delta_n) and has only finitely many integer points. $\endgroup$
    – Alison Miller
    Commented Jan 23, 2023 at 2:29
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    $\begingroup$ Apparently the missing assumption is that there is a real direction on which $F_{2n}=0$. Otherwise it is strange, indeed. $\endgroup$
    – fedja
    Commented Jan 24, 2023 at 0:33

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