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Are there general non-trivial conditions on $p(\cdot)$ and $n$, where $p(\cdot)$ is a product of cyclotomic polynomials and $n$ is a positive integer, such that all the coefficients of $p(\cdot)$ are nonnegative and such that $p(z)$ is rational (and hence a rational integer) whenever $z$ is an $n$th root of 1?

Two examples should clarify the question: $p(z) = \Phi_5(z)$ $=$ $1 + z + z^2 + z^3 + z^4$ is rational whenever $z$ is a 4th or 5th root of 1, while $p(z) = \Phi_5(z) \Phi_6(z)$ $=$ $1 + z^2 + z^3 + z^4 + z^6$ is rational whenever $z$ is a 5th or 6th root of 1.

I imagine that there are general conditions (some trivial and some less so) under which rationality is guaranteed, as well as sporadic cases under which rationality holds "by accident".

Ideally I would like not just a criterion but a generic construction (applying whenever the criterion holds) of a permutation $\pi$ of some set of cardinality of $p(1)$ such that for all integers $k$ the number of fixed points of $\pi^k$ equals the magnitude of $p(\zeta^k)$ where $\zeta$ is a primitive $n$th root of 1; or, failing that, an invertible linear map $A$ from some $p(1)$-dimensional vector space to itself such that for all integers $k$ the trace of $A^k$ equals $p(\zeta^k)$. (E.g., for the four pairs $p(\cdot),n$ mentioned above, the permutations $\pi$ should have respective cycle-types (1)(2345), (12345), (12345), and (12)(345).)

Such a construction would be relevant to the Galois-theoretic perspective on the cyclic sieving phenomenon. (The CSP was introduced by Reiner, Stanton, and White in 2004 and is discussed for instance in http://arxiv.org/abs/1008.0790 and http://www.ams.org/notices/201402/rnoti-p169.pdf .)

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  • $\begingroup$ To see why I included the condition that the coefficients of $p(\cdot)$ are nonnegative, consider the case $p(z)$ $=$ $\Phi_6(z)$ $=$ $1-z+z^2$, $n=2$; we have $p(1)=1$ and $p(-1)=3$, which is incompatible with the existence of a corresponding permutation $\pi$ or linear map $A$. $\endgroup$ Dec 28, 2014 at 17:47
  • $\begingroup$ A somewhat trivial example is associated to all primes $p\equiv 1\pmod n$. The corresponding cyclotomic polynomial evaluates to $p$ at $1$ and to $1$ at all other $n$-th roots of $1$. A corresponding permutation has a unique fixpoint and $(p-1)/n$ cycles of length $1$. An interesting observation is the fact that all (with or without the positivity condition) such polynomials (for a given value of $n$) form a multiplicative monoid. This gives many more polynomials (I do not know if the monoid construction can be lifted to suitable permutations). $\endgroup$ Dec 29, 2014 at 15:53
  • $\begingroup$ In fact, concerning my comment above, all polynomials of the form $(1-x^{kn+1})/(x-1)$ work, even for $kn+1$ not a prime. $\endgroup$ Dec 29, 2014 at 17:19
  • $\begingroup$ Where Roland writes "cycles of length 1" (in his first comment), I assume he means "cycles of length $n$". However I do not understand the second half of that comment (partly because the meaning of "such polynomials" is unclear to me). I note that if $q(z)$ and $r(z)$ are polynomials that evaluate to $p$ at $z=1$, then $q(z)r(z)$ (the most obvious monoid product to consider) has the value $p^2$, not $p$, at $z=1$. So I think I must be off-base in my effort to grasp what Roland is saying. $\endgroup$ Dec 30, 2014 at 3:21
  • $\begingroup$ No, this is what I meant: the obvious product for polynomials. $\endgroup$ Dec 30, 2014 at 11:59

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