This is something that came out of working on a problem:

Let $m$ be an odd positive integer and $f \in \mathbb Q[x]$ be a polynomial of degree less than $m$. With $\zeta_m$ denoting a primitive root of unity, suppose that $|f(\zeta_m)|=1$ but $f(\zeta_m)$ is not a root of unity. Find the minimum value of the product $$P_{f, m}(n):= \prod_{a \in (\mathbb Z / m \mathbb Z)^\times} |f(\zeta_m^a)^{2^{n+1}} - f(\zeta_m^{2a})^{2^n}|$$ over all natural numbers $n$. I would like to know

  1. The minimum $\min_{n \in \mathbb N} P_{f, m}(n)$ possibly as a function of $m$ and (perhaps) the coefficients of $f$. For which $n$ (of course, allowed to depend on $m$, $f$) is this bound attained?

  2. If there is any "absolute" lower bound on the product $P_{f, m}(n)$ over all natural numbers $n$ and over all polynomials $f \in \mathbb Q[x]$ with $\deg f \leq m-1$ which is positive (and in terms of $m$). For which $n$ and $f$ (of course, allowed to depend on $m$) is this bound attained?

  3. (Perhaps too much to hope) Can we describe the behaviour of the function $P_{f, m}(n)$ as a function in $n$, again depending on the given polynomial $f$ and $m$?

Since all the Galois conjugates of $f(\zeta_m)$ are given by $\{f(\zeta):\zeta \in \mu_m\}$ (where $\mu_m$ denotes the group of the primitive $m$-th roots of unity) and the extension $\mathbb Q(\zeta_m)/\mathbb Q$ is abelian so that complex conjugation commutes with all other elements of the Galois group, I could reformulate the given problem as the following:

Let $m \in \mathbb N$, $f \in \mathbb Q[x]$ with $\deg f \leq m-1$ such that $f(\zeta) \in \mathbb{S}^1 \setminus (\bigcup_{r \in \mathbb N} \mu_r)$ for all $\zeta \in \mu_m$. Find the respective minimum values (as above) of the product $P_{f,m}(n) = \prod_{\zeta \in \mu_m} |f(\zeta^a)^{2^{n+1}} - f(\zeta^{2a})^{2^n}|$.

Apart from this, I tried explicitly writing out $f$ and the above product in terms of a sequence of polynomials but (as expected I guess), that got pretty out of hand pretty quickly. I would really appreciate any suggestions. I'm not sure if this would help but we may also assume that $m$ is odd (in all of the above problems).

P.S.: The exact function that I am trying to find the upper bound (over all natural numbers $n$) of (again in terms of $m$, $f$, both or neither) is actually $$G_{f,m}(n):= \frac{1}{2^n} \log\frac{2^{n+1}}{P_{f,m}(n)^{1/\phi(m)}} = \frac{1}{2^n} \left( (n+1) \log 2 - \frac{1}{\phi(m)} \log P_{f, m}(n) \right)$$ so if it is easier to directly find the corresponding upper bounds of $G_{f, m}(n)$ then I would really appreciate some suggestions for the same.



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