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Let's call a polynomial $P(x)\in \mathbb Z[x]$ realizable if there exists some polynomial-count scheme $X$ over $\mathbb Z$ that satisfies $|X(\mathbb F_q)|=P(q)$, for all prime powers $q$.

Suppose $n$ is a positive integer that is not a prime power. Can one always find a realizable polynomial that satisfies $P(n)<0$?

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    $\begingroup$ Do you know of such a polynomial for say, $n=6$? $\endgroup$
    – dhy
    Commented Nov 25, 2020 at 0:32
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    $\begingroup$ @dhy There is a combinatorial construction in D. Glynn, "Rings of geometries II" J. Combin. Theory Ser. A 49 (1988), no. 1, 26-66. The author essentially shows that a certain polynomial of degree 14 gives a negative value at 6, but can be interpreted as counting the points of some variety. This variety is defined combinatorially, in a way that would allow you to define its points over a finite projective plane. The author uses this to prove the nonexistence of finite planes of order 6. Of course, no such example is known for $n>6$. $\endgroup$
    – Gjergji Zaimi
    Commented Nov 25, 2020 at 1:36
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    $\begingroup$ I found a more modern reference that has essentially the same example, written in geometric language. See table 1 in the paper arxiv.org/abs/2007.16014 $\endgroup$
    – Gjergji Zaimi
    Commented Nov 25, 2020 at 7:32

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