# Minimum of a product of polynomial evaluated at primitive roots of unity, given that the value of the polynomial at the same lies on unit circle

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.