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### Which sets of roots of unity give a polynomial with nonnegative coefficients?

The question in brief: When does a subset $S$ of the complex $n$th roots of unity have the property that
$$\prod_{\alpha\, \in \,S} (z-\alpha)$$
gives a polynomial in $\mathbb R[z]$ with ...

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### Independent units in pure number fields $\mathbb{Q}(\sqrt[p]{t})$

Theorem: In this paper of Frei and Levesque, they correct the proof of a result of Halter-Koch and Stender:
Define the real pure algebraic number field $\mathbb{K}=\mathbb{Q}(\omega_n)$ for ...

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### Rational viewing points in a polygon

We refer to the question posed in Seeing the vertices of a polygon with rational angles, but now ask for constructions or for the existence of rational viewing points. We'll call a point $p$ inside ...

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### Condensation points of orbits of roots of unity

For a fixed $n\in \mathbb{N}$ we consider the set of $n$-roots of unity $R(n)=\{z\in S^1; z^n=1\}$. It splits into mutually disjoint orbits. Let $R=\bigcup_{n=0}^{\infty} R(2^n-1)$. For each orbit in ...

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321 views

### Looking for product of symmetric polynomials evaluated at roots of unity

Consider $a_{1} = \alpha^{N-n}, a_{2} = \alpha^{N-n+1}, a_{3} = \alpha^{N-n+2}, a_{4} = \alpha^{N-n+3}, \cdots, a_{n} = \alpha^{N-1}$ where $\alpha$ is a complex $N$th root of unity where $N = 2 + ...