Questions tagged [roots-of-unity]
The roots-of-unity tag has no usage guidance.
27
questions with no upvoted or accepted answers
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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 ...
- 1,513
8
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347
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Computing coefficients of polynomials from roots in $O(n\log{n})$ time
Suppose I have a univariate polynomial $p$ over a prime-order finite field $\mathbb{F}_q$ whose roots I know.
Suppose that the roots of $p$ are always an $n$-sized subset of $R=\{1,2,\dots,N\}, N <...
- 191
7
votes
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147
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Finding when a certain product in a cyclotomic field is equal to one
For the following, fix $m\geq 2$, and let $a_{0},\dots,a_{m-1}\in\mathbb{Z}$ be such that $\sum_{k=0}^{m-1}a_{k}=0$. I would like to find the exact conditions on the $a_{k}$ so that the following ...
- 299
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Simultaneous vanishing $\mathbb{Q}$-linear relations between $N$-th roots of unity
Let $\zeta$ be a primitive $N$-th root of unity and $\Gamma \subset (\mathbb{Z}/N)^\times$ a subgroup. Let $|\Gamma|$ be the cardinality of $\Gamma$ and consider the linear map $M_\Gamma\colon \mathbb{...
- 171
6
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224
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Infinitude of cyclotomic polynomials with a certain number of terms
Let $\Phi_n$ be the $n$th cyclotomic polynomial:
$${\Phi _{n}(x)=\!\!\prod _{\substack {1\leq k\leq n \\ \gcd(k,n)=1}} \!\!\big(x-e^{2i\pi {k/n}}\big).}$$
Here is a list of the first 30 cyclotomic ...
- 615
5
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272
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Is an algebraic number satisfying certain super-congruences a root of unity?
Let $K|\mathbb{Q}$ be a number field, $D$ its discriminant and $\mathcal{O}$ the ring of integers in $K$. Let $x\in K$ (or maybe $\in \mathcal{O}[\frac 1D]$) such that for all primes $p$ in $\mathbb{Q}...
- 171
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119
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Sign preserving Galois automorphisms
I have an algebraic number $\alpha \in \mathbb{Q}(\zeta)$, where $\zeta^n = 1$ is a root of unity (not primitive) given as a linear combination of powers of $\zeta$, i.e, $\alpha = \sum_{i=1}^k a_i \...
- 305
4
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127
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Vanishing exponential sums of fractional parts of polynomials
Let $p$ be an integer polynomial and $k$ be a natural number, both fixed. Is it the case that if
$$C(\alpha) = \sum_{i=1}^k e(\alpha \{p(i)/k\})$$
equals 0, then $\alpha$ is an integer? Here, $e(x) = ...
- 41
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289
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power series and roots of unity
Let $p$ be an odd prime and $X$ and $Y$ be subsets of $p^{th}$ roots of unity, $|X|=|Y|=n,X\neq Y.$ Let $f(t)=\sum_{x\in X}x^{t}-\sum_{y\in Y}y^{t}$. If $f(t)=at^k+o(t^k)$ is the power series ...
- 205
4
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146
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An Optimization Problem for Exponential Polynomials
Let $\omega$ be a primitive complex $n^{th}$ root of unity. I am interested in the following quantity
$$
\max_{f(n)\leq \ell \leq g(n)} \quad \min_{0<k\leq n-1}
\left| 1+\omega^k+\omega^{2k}+\...
- 10.1k
4
votes
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268
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How small can the nonzero sum of $O(\log n)$ distinct $n-$th roots of unity be?
The OEIS sequence oeis.org/A108380 gives the least number of distinct n-th roots of unity summing to the smallest possible nonzero magnitude.
This sequence seems to imply that the least number of ...
- 10.1k
4
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339
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Independent units in pure number fields $\mathbb{Q}(\sqrt[p]{t})$
Theorem: In this paper of Frei and Levesque, they correct the proof of a result of Halter-Koch and Stender:
Define the real pure algebraic number field $\mathbb{K}=\mathbb{Q}(\omega_n)$ for $\omega_n=...
- 1,985
3
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114
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Third roots of unity and norm element
Let $K = \mathbb{Q}(\zeta_3)$ where $\zeta_3$ is a third root of unity, let $F = \mathbb{Q}(\sqrt[3]{\ell})$ where $\ell \equiv 1 \pmod{9}$ is a prime and set $L$ to be the Galois closure of $F$, i.e.,...
- 1,191
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Coefficients for Expansions of $1-\zeta_p$
Let $\mathbb{Q}_p(\zeta_p)$ be the cyclotomic extension of the $p$-adic field $\mathbb{Q}_p$. Then $1 - \zeta_p$ is a uniformizer for this field. Recall that
$$\sum_{i=1}^{p-1} \zeta_p^i = -1.$$
So ...
- 31
3
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160
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Modular root of $-1$
Take two distinct coprime integers $a,b$ and take $q=a^2+b^2$. Consider the equation $(xy^{-1})^2\equiv-1\bmod q$. The solutions are two roots which correspond to $(x,y)=(a,b)$ and $(x,y)=(b,a)$. Look ...
- 13.7k
3
votes
1
answer
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Homomorphism from integral module generated by roots of unity to cyclic group?
Let $S$ be the set of all roots of unity. Consider the $\mathbb{Z}$-module, $\mathbb{Z} S$, as an additive abelian group (that is, $\mathbb{Z} S$ is the subset of complex numbers that can be ...
- 61
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Sum of roots of unity
Today I came across the series $\sum_{k=0}^{n-1}\varepsilon^{2k^2}$, where $\varepsilon$ is some primitive $n^\text{th}$ root of unity. Is there an explicit expression for this sum? I mistakenly ...
- 8,076
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225
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Finite sum involving root of unity
I have the following sum:
$$\sum_{\sigma=1,\text{odd}}^{\frac{p}{2}-2}\frac{\sin \frac{r \sigma \pi}{p}}{\sin \frac{q \sigma \pi}{p}}$$
where $p\equiv2\pmod4$, $p$ and $q$ are coprime numbers such ...
- 21
2
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372
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Trace of roots of unity has valuation more than 1 in uramified field
Let $F$ be a finite extension of $\mathbb{Q}_p$ (p is prime) and $K/F$ be a unramified extension of prime degree $\ell (\neq p)$. Denote $\mu_K$ be the group of roots of unity in $K.$ Does there exist ...
- 255
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112
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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 (...
- 613
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41
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About nilpotent Jordan algebras, matrix representations and formally real algebras
Given an non-commutative associative unital algebra A of characteristic $0$, one can construct a Jordan algebra $A+$ using the same underlying addition vector space.
Notice first that an associative ...
- 733
1
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120
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$\sin(\frac{\pi}{p}) $ not expressible by positive radicals and $\sin(\frac{\pi}{q_i})$?
We have the following identities:
$\sin(\frac{\pi}{1})=0$
$\sin(\frac{\pi}{2})=1$
$\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$
$\sin(\frac{\pi}{4})=\sqrt{\frac{1}{2}}$
Lets start with a definition.
Rules ...
- 733
1
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90
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Mod $N^2$ evaluation of a polynomial defined by first $N-1$ roots
Given a prime $N$ and integer $g$, where $g$ is able to generate the multiplicative subgroup $(\mathbb{Z}/N^2\mathbb{Z})^*$, I am interested in any results simplifying or evaluating $f\in (\mathbb{Z}/...
- 11
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130
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How to evaluate this sum of roots of unity with condition to zero
In evaluating the sum:
$$\tag{1}\label{e1}\sum\limits_{j = 1 < h}^N {\left( {{e^{\frac{{i\pi }}{N}\left( {{s_1}j + {s_2}h} \right)}} - {e^{\frac{{i\pi }}{N}\left( {{s_1}h + {s_2}j} \right)}}} \...
- 59
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206
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Discrete Fourier transform of the Ramanujan's sums
Let $n$ be a positive integer, and $\zeta_n$ a primitive $n$-root of unity.
I consider the polynomial $P(X) = \sum_{k=0}^{\phi(n)-1} \left[ \sum_{l \in \mathbb{Z}_n^*}^n \zeta_n^{kl} \right]X^k = \...
- 313
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90
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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 $...
0
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662
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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 + (n-...
- 13.7k