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Given $f(x_1,\dots,x_n)\in\mathbb Z[x_1,\dots,x_n]$ such that

  1. Each coefficient is bound in absolute value by $B$

  2. Degree of each variable in any monomial is bound by $d$

  3. Total degree is $d'$

  4. $f(x_1,\dots,x_n)$ is irreducible in $\mathbb Z[x_1,\dots,x_n]$

then what is the maximum number integral roots $(y_1,\dots,y_n)\in\mathbb Z^n$ it can have such that in a $\ell$ length cube?

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  • $\begingroup$ it's not clear where your hypercube is. Do you mean the convex closure of $(\pm\ell, \pm\ell,...,\pm\ell)$, or something else? $\endgroup$ – Dima Pasechnik Jan 9 at 13:19
  • $\begingroup$ @DimaPasechnik That is the cube. $\endgroup$ – Freeman. Jan 9 at 13:41
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    $\begingroup$ For constant $d'$, it is easy to prove a bound of the form $(2\ell)^{n-1}(1+o(1))$, which is sharp in view of the equation $x_1-x_2=0$. In fact, this is true even modulo a prime (Lang--Weil theorem). I suppose that the question is to get good dependence as $d'\to\infty$, right? $\endgroup$ – Boris Bukh Jan 9 at 14:02

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