Take $ 1<d_1,\dots,d_t<n$ where $ n=2^kp^rq^m$ , $ d_i|n$ and $ n-\sum_{i=1}^t\varphi(d_i)=\Theta((\log n)^{2e})$ at some $ e\geq1$ on condition vector $ \pi_{c}(\prod_{i=1}^t\Phi_{d_i}(x))$ which is $ c$ right cyclic shift of vector $ $ \begin{bmatrix}a_0&a_1&\dots&a_{\sum_{i=1}^t\varphi(d_i)-1}&a_{\sum_{i=1}^t\varphi(d_i)}\begin{matrix}0&\dots&0\end{matrix}\end{bmatrix}\in\Bbb Z^{2^{k}p^{r}q^{m}}$ $ (we have $ n-\big(1+deg\big(\prod_{i=1}^t\Phi_{d_i}(x)\big)\big)=n-(1+\sum_{i=1}^t\varphi(d_i))$ padded zeros) has the property that $ $ \pi_{c}\big(\prod_{i=1}^t\Phi_{d_i}(x)\big)=\pi_{c’}\big(\prod_{i=1}^t\Phi_{d_i}(x)\big)\iff c\equiv c’\bmod \big(1+deg\big(\prod_{i=1}^t\Phi_{d_i}(x)\big)\big).$ $

Denote highest absolute value of coefficients of $ \prod_{i=1}^t\Phi_{d_i}(x)$ by $ \Phi_{\max}$ .

If $ \mathcal R_n=\{e^{\frac{2\pi \cdot i\cdot\sqrt{-1}}n}:0\leq i<n\}$ then what is the distribution of $ \max_{\zeta\in \mathcal R_n}|\prod_{i=1}^t\Phi_{d_i}(\zeta)|$ ?

If such $ \prod_{i=1}^t\Phi_{d_i}(x)$ exists then is it possible $ \max_{\zeta\in \mathcal R_n}|\prod_{i=1}^t\Phi_{d_i}(\zeta)|<\frac{n\cdot\Phi_{\max}^{1/2}}{2^{O((\log n)^e)}}$ holds because of cancellations.

Is there any examples of such massive cancellations in cyclotomic polynomials?

Note what I seek might not be unlikely since roots of each of $ \Phi_{d_i}(x)$ lie in $ \mathcal R_n$ and if the polynomial wanders off arbitrarily at other relevant roots of unity it might be its degree of $ \sum_{i=1}^t\varphi(d_i)$ is smaller than needed. This may keep the $ \max_{\zeta\in \mathcal R_n}|\prod_{i=1}^t\Phi_{d_i}(\zeta)|$ distribution in a tight range.