Maximum number of integral roots in degree $d$ polynomial?

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?

• it's not clear where your hypercube is. Do you mean the convex closure of $(\pm\ell, \pm\ell,...,\pm\ell)$, or something else? – Dima Pasechnik Jan 9 at 13:19
• @DimaPasechnik That is the cube. – Freeman. Jan 9 at 13:41
• 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? – Boris Bukh Jan 9 at 14:02