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?

  • $\begingroup$ 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
  • 3
    $\begingroup$ 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

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.