Computing a polynomial product over roots of unity I'm trying to compute the coefficients of the following polynomial, where $\omega$ is a primitive $p$-th root of unity, for $p$ prime:
$$a(x) = \prod_{i=0}^{p-1} f(\omega^ix).$$
It turns out that the $i$-th coefficient is always an integer, and non-zero only when $i$ is a multiple of $p$. So it seems to me like there should be an elementary expression for $a$.
So far I've got this expression for the $i$-th coefficient:
$$a_i = x^i\sum_{k_0 + \ldots + k_{p-1} = i}f_{k_0} \cdots f_{k_{p-1}} \omega^{k_1 + 2k_2 + \ldots + (p-1)k_{p-1}}$$
where each $k_i$ is non-negative and bounded by the degree of $f$.
Clearly the roots of unity all cancel out somehow, but I can't figure out how to get a 'nice' expression out. Any suggestions?
 A: The transformation
$$
  \mathcal{L}_n : \mathbb{C}(x) \longrightarrow \mathbb{C}(x) 
$$
defined by
$$
  \mathcal{L}_n(F(x)) = \sum_{\zeta\in\boldsymbol\mu_n} F(\zeta y)\Big|_{y^n\to x}
$$
is called a (generalized) Landen transformation. If you take $F(x)=1/f(x)$, where $f(x)$ is a polynomial, then the denominator of $\mathcal{L}_n(F(x))$ is more-or-less the product you are studying. See http://arxiv.org/abs/1308.5355 (especially Section 8, which is really about $\prod_{\zeta\in\boldsymbol\mu_n} F(\zeta y)\Big|_{y^n\to x}$), as well as the references in that paper.
A: A slightly more general result is shown in Theorem A here
https://www.maths.tcd.ie/pub/ims/bull47/R4701.pdf
which states:
If $r$ is relatively prime to $n$ then the coefficient of $x^r$ in $\prod_{i=1}^n f(\omega^i x)$ is 0.
(Here $\omega$ is a primitive $n$-th root of unity, $f$ has rational coefficients.)
A: Notice that
$$
y^p-x^p=\prod_{i=0}^{p-1} (y-\omega^i x)\ .
$$
Therefore
$$
\prod_{i=0}^{p-1} f(x\omega^i)= Res_y (y^p-x^p, f(y))\ .
$$
Where $Res_y$ is the resultant of the polynomials in $y$. The above is just a particular case of the so called Poisson product formula. You can then compute the resultant using for instance Sylvester's determinant formula. 
See the review
http://mate.dm.uba.ar/~alidick/papers/chapter1cd.pdf
by Cattani and Dickenstein
for a nice introduction to resultants and their properties.
