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Given $m,n\in\Bbb Z_{>1}$ is there a way to quantify smallest degree (when exists) $f(x)\in\Bbb Z_m[x]$ with $f(q)\bmod m\equiv1\iff q<n\mbox{ is prime }$ and is $deg(f(x))=O(\log(mn))$?

Prime detection has a polynomial size circuit because of the existence of a polynomial time algorithm.

The open problem here is 'Can prime detection have a polynomial size formula?'.

$m=O(n^c)$ and $deg(f(x))=O((\log(mn))^c)$ will give a polynomial size formula and $c=1$ gives a linear size formula.

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migrated from mathoverflow.net Nov 17 '17 at 4:36

This question came from our site for professional mathematicians.

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    $\begingroup$ Surely if $m=n$ and $n$ is prime then the minimum degree is $\pi(n-1)\sim n/\log(n)$. $\endgroup$ – Neil Strickland Oct 27 '17 at 9:52
  • $\begingroup$ @NeilStrickland Sure bound is true? $\endgroup$ – T.... Oct 27 '17 at 15:48
  • $\begingroup$ What you state looks like upper bound. Why does the minimum degree have to follow prime number theorem? $\endgroup$ – T.... Oct 28 '17 at 10:39
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If $m$ is a prime, then the integers mod $m$ form a field and the fundamental theorem of algebra holds: to have $\pi(n-1)$ roots, you need to have degree $\pi(n-1)$. So $O(\log(mn))$ cannot hold in general.

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  • $\begingroup$ actually degree $\pi(n-1)$ also can have poly size formula if polynomial is sparse or has a highly non-trivial internal structure. $\endgroup$ – T.... Nov 12 '17 at 22:25
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    $\begingroup$ @Turbo You changed the question with your recent edits. I was addressing only your original question. If you'd like an answer to the new question, it should be posted as a separate question. $\endgroup$ – Charles Nov 12 '17 at 22:33
  • $\begingroup$ a poly size formula? $\endgroup$ – T.... Nov 12 '17 at 22:48

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