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Take a bivariate polynomial of degree $d_x+d_y>\max(d_x,d_y)>1$ in $\mathbb Z[x,y]$ with coefficients bound in absolute value by $b$ ($d_x$ is $x$-degree and $d_y$ is $y$-degree).

  1. What is the maximum number of integer roots it can have in the box $[-t,t]\times[-t,t]$ for fixed $d_x$ and $d_y$?

I am looking for dependence on $b,t$ for fixed $d_x,d_y$ and possibly extremal examples.

Are maximal family of examples known in literature or can be constructed?

  1. What is the distribution of number of integer solutions if the solutions are restricted to the box?
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    $\begingroup$ Let $f(x,y)$ be your bivariate polynomial. By a result of Bombieri and Pila from 1989, one has that for any $\epsilon > 0$ there exists a number $c_{d,\epsilon}$ depending on $d = d_x + d_y$ and $\epsilon$ but not on $b, t$ such that the number of solutions to $f(x,y) = 0$ in $[-t,t]^2$ is at most $c_{d,\epsilon} t^{1/d + \epsilon}$. This is essentially sharp, as one can almost reach this bound by taking something like $f(x,y) = y - x^d$. $\endgroup$
    – Stanley Yao Xiao
    Commented Nov 3, 2020 at 20:54
  • $\begingroup$ I think in your example d is max degree not sum and nevertheless it is helpful. $\endgroup$
    – Turbo
    Commented Nov 3, 2020 at 21:42

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