Questions tagged [roots-of-unity]
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80 questions
0 votes
0 answers
143 views
Identity involving primitive roots of unity
Let $m,d\in\mathbb N$ with $\gcd(m,d)=1$. Write $\xi_m=e^{2\pi i /m}$, a primitive $m$-th root of unity. Let $r_1,r_2$ be integers with order exactly $d$ in $\mathbb Z_m^\times$ (i.e. $r_i^d\equiv 1\...
- 2,589
7 votes
3 answers
1k views
Peculiar exception in the number of distinct values taken by the sums of the 6th degree roots of unity
For a nonnegative integer $n$, let $N_n$ be the number of distinct values taken by the sums of $n$ 6th-degree roots of unity (with repetitions). First few counts are $N_0=1$, $N_1=6$, $N_2=19$, and ...
- 38.3k
0 votes
0 answers
143 views
Probabilistic interpretation of roots of unity in $\mathbb{C}^2$
I have a question that concerns how often a special class of bivariate polynomials (which I will call mask polynomials) intersects the set of roots of unity in $\mathbb{C}^2$. Caveat: I consider ...
1 vote
0 answers
107 views
How to find a root of unity satsifying the following equation for tate/ate pairing inversion?
The aim is for pairing inversion where miller inversion can only work if an equation is satisfied. So given a finite field modulus $q$ having degree $k$ ; and a finite field element $z$ having ...
- 67
2 votes
0 answers
129 views
Discrete Fourier Transform using not the n-th roots of unity, but the 2n-th primitive roots of unity
I should start with stating that my question stems from the proof of lemma 6 in the following paper: Jung Hee Cheon, Hyeongmin Choe, Julien Devevey, Tim Güneysu, Dongyeon Hong, Markus Krausz, Georg ...
- 21
0 votes
0 answers
136 views
Some invariant subsets of roots of unity
Let $n$ and $t$ be positive integers with $\gcd(t,n)=1$, let $C$ be the collection of all primitive $n$th roots of unity and let $S$ be a subset of $C$ such that, for every $\ell\in\mathbb{N}$, $$(1)\...
- 1,116
0 votes
0 answers
111 views
Points on a circle related by involution mapping
Related to but different from Points on a circle with near-zero centroid I have the following puzzle: Let's assume a set of points $\vec{a} = \{a_1, a_2, \ldots, a_n\}$, with $a_i$ being a real number ...
- 109
0 votes
1 answer
239 views
Points on a circle with near-zero centroid
I want to find sets of $N$ unit complex numbers $z_j = \exp(\rm{i}\phi_j)$ whose mean is close to zero, i.e., $c = \frac{1}{N}\sum_{j=1}^N z_j; |c|\leq t$, where $t\ll1$ is some threshold. By unique, ...
- 109
1 vote
0 answers
122 views
Period of the modulus of a complex exponential sum
Consider a sum of exponentials function of the integer $x$: $f(x)=A(x)+B(x)$, where $A(x)=\sum_{i=1}^n c_i \theta_i^x$ with $\theta_i$ roots of unity, and $B(x)=\sum_{j=1}^m d_j\lambda_j^x$ with $|...
- 373
2 votes
0 answers
206 views
Monomial symmetric polynomials evaluation at roots of unity
The monomial symmetric polynomials are defined see Wikipedia. For an arbitrary partition $\lambda$ with $n$ parts I'm trying to find the following values: $$m_{\lambda}(\omega_0,\dotsc,\omega_{n-1})$$ ...
- 63
4 votes
1 answer
291 views
Third roots of unity and norm element
Let $K = \mathbb{Q}(\zeta_3)$ where $\zeta_3$ is a third root of unity, let $F = \mathbb{Q}(\sqrt[3]{\ell})$ where $\ell \equiv 1 \pmod{9}$ is a prime and set $L$ to be the Galois closure of $F$, i.e.,...
- 1,353
1 vote
0 answers
71 views
Finding the radical expressions of trig functions [closed]
I am trying to find the exact radicals of the sine and cosine of (m/n)*pi. I have been using sympy to do this and made this pretty good script: ...
4 votes
0 answers
200 views
Vanishing exponential sums of fractional parts of polynomials
Let $p$ be an integer polynomial and $k$ be a natural number, both fixed. Is it the case that if $$C(\alpha) = \sum_{i=1}^k e(\alpha \{p(i)/k\})$$ equals 0, then $\alpha$ is an integer? Here, $e(x) = ...
- 41
2 votes
0 answers
330 views
Sum of roots of unity
Today I came across the series $\sum_{k=0}^{n-1}\varepsilon^{2k^2}$, where $\varepsilon$ is some primitive $n^\text{th}$ root of unity. Is there an explicit expression for this sum? I mistakenly ...
- 12.9k
1 vote
1 answer
216 views
Norm of $2^{i}$-th primitive root
Let $ K $ be finite degree extension of $ \mathbb{Q} $ such that $ -1 $ is not a square in $ K $. Let $ L = \frac{K[x]}{\langle x^2 +1\rangle}$. Thus every element of $ L $ is of the form $ a + ib $ ...
- 923
1 vote
0 answers
94 views
About nilpotent Jordan algebras, matrix representations and formally real algebras
Given an non-commutative associative unital algebra A of characteristic $0$, one can construct a Jordan algebra $A+$ using the same underlying addition vector space. Notice first that an associative ...
- 799
1 vote
0 answers
152 views
$\sin(\frac{\pi}{p}) $ not expressible by positive radicals and $\sin(\frac{\pi}{q_i})$?
We have the following identities: $\sin(\frac{\pi}{1})=0$ $\sin(\frac{\pi}{2})=1$ $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$ $\sin(\frac{\pi}{4})=\sqrt{\frac{1}{2}}$ Lets start with a definition. Rules ...
- 799
3 votes
1 answer
412 views
A similar relationship between the generic cubic and the Lehmer quintic?
I. Comparison It doesn't seem to be well-known that the generic cubic (prominent in this MO post) for $C_3 = A_3$, $$x^3-nx^2+(n-3)x+1 = 0$$ has the nice property that its roots $a,b,c$, if in correct ...
- 13k
5 votes
1 answer
364 views
A conjectural permanent identity
Let $n>1$ be an integer, and let $\zeta$ be a primitive $n$th root of unity. By $(3.4)$ of arXiv:2206.02589, $1$ and those $n+1-2s\ (s=1,\ldots,n-1)$ are all the eigenvalues of the matrix $M=[m_{jk}...
- 17.6k
2 votes
1 answer
572 views
Integer eigenvalues of a class of matrices inspired by Prof. Zhi-Wei Sun's conjecture
Theorem: Let $n>1$ be an odd number and $\zeta$ a primitive $n$-th root of unity. Then \begin{eqnarray} &&\sum_{\tau\in D(n-1)}\mathrm{sign}(\tau)\prod_{j=1}^{n-1}\frac{1}{1-\zeta^{j-\tau(j)...
4 votes
1 answer
622 views
Has any one seen this sum of roots of unity before?
Fix a prime $p >2$ and $q_1$, $q_2$ such that $q_i - 1$ is exactly divisible by $p$. For any $n$, $a$, $b $, consider the sum $$\sum_{i=0}^{p^{n-1}-1}\zeta_{p^n}^{aq_1^i+bq_2^i}.$$ Is this always ...
- 8,081
7 votes
0 answers
186 views
Finding when a certain product in a cyclotomic field is equal to one
For the following, fix $m\geq 2$, and let $a_{0},\dots,a_{m-1}\in\mathbb{Z}$ be such that $\sum_{k=0}^{m-1}a_{k}=0$. I would like to find the exact conditions on the $a_{k}$ so that the following ...
- 299
15 votes
2 answers
1k views
Vanishing of a sum of roots of unity
In my answer to this question, there appears the following sub-question about a sum of roots of unity. Denoting $z=\exp\frac{i\pi}N$ (so that $z^N=-1$), can the quantity $$\sum_{k=0}^{N-1}z^{2k^2+k}$$ ...
- 53.1k
3 votes
0 answers
221 views
Coefficients for Expansions of $1-\zeta_p$
Let $\mathbb{Q}_p(\zeta_p)$ be the cyclotomic extension of the $p$-adic field $\mathbb{Q}_p$. Then $1 - \zeta_p$ is a uniformizer for this field. Recall that $$\sum_{i=1}^{p-1} \zeta_p^i = -1.$$ So ...
- 31
1 vote
1 answer
260 views
A determinant involving the cotangent function
Let $n>1$ be odd. In my 2019 preprint On some determinants involving the tangent function, I proved that $$\det\left[\tan\pi\frac{aj+bk}n\right]_{1\le j,k\le n-1}=\left(\frac{-ab}n\right)n^{n-2}$$ ...
- 17.6k
10 votes
1 answer
649 views
Identities involving derangements and roots of unity
For a positive integer $n$, a derangement of $\{1,\ldots,n\}$ is a permutation $\tau$ of $\{1,\ldots,n\}$ with $\tau(j)\not=j$ for all $j=1,\ldots,n$. For convenience, we let $D(n)$ denote the set of ...
- 17.6k
11 votes
1 answer
724 views
A conjecture on binomial coefficients and roots of unity
Is the following true? Let $p$ be a prime and let $w$ be a $(p-1)$st root of unity (not necessarily primitive). Then $$\binom{w}{n}=\frac{w(w-1)\cdots(w-n+1)}{n!}$$ is $p$-integral; i.e., it can be ...
- 17.7k
1 vote
0 answers
112 views
Mod $N^2$ evaluation of a polynomial defined by first $N-1$ roots
Given a prime $N$ and integer $g$, where $g$ is able to generate the multiplicative subgroup $(\mathbb{Z}/N^2\mathbb{Z})^*$, I am interested in any results simplifying or evaluating $f\in (\mathbb{Z}/...
- 11
6 votes
0 answers
150 views
Simultaneous vanishing $\mathbb{Q}$-linear relations between $N$-th roots of unity
Let $\zeta$ be a primitive $N$-th root of unity and $\Gamma \subset (\mathbb{Z}/N)^\times$ a subgroup. Let $|\Gamma|$ be the cardinality of $\Gamma$ and consider the linear map $M_\Gamma\colon \mathbb{...
- 171
2 votes
1 answer
402 views
Möbius inversion formula and roots of unity
Is the exact value of $$ \sum_{d\mid n} \mu\left(\frac{n}{d}\right) \zeta^d $$ known? Here, $\mu$ denotes the Möbius function and $\zeta$ a root of unity. At first sight, it seems to me that this ...
- 171
8 votes
0 answers
425 views
Computing coefficients of polynomials from roots in $O(n\log{n})$ time
Suppose I have a univariate polynomial $p$ over a prime-order finite field $\mathbb{F}_q$ whose roots I know. Suppose that the roots of $p$ are always an $n$-sized subset of $R=\{1,2,\dots,N\}, N <...
- 201
2 votes
0 answers
258 views
Finite sum involving root of unity
I have the following sum: $$\sum_{\sigma=1,\text{odd}}^{\frac{p}{2}-2}\frac{\sin \frac{r \sigma \pi}{p}}{\sin \frac{q \sigma \pi}{p}}$$ where $p\equiv2\pmod4$, $p$ and $q$ are coprime numbers such ...
- 21
5 votes
0 answers
293 views
Is an algebraic number satisfying certain super-congruences a root of unity?
Let $K|\mathbb{Q}$ be a number field, $D$ its discriminant and $\mathcal{O}$ the ring of integers in $K$. Let $x\in K$ (or maybe $\in \mathcal{O}[\frac 1D]$) such that for all primes $p$ in $\mathbb{Q}...
- 171
6 votes
2 answers
576 views
A conjecture involving roots of unity
Motivated by Kevin Liu's recent question, here I pose the following conjecture based on my numerical computation. Conjecture. Let $m>1$ and $n>1$ be integers. Let $\delta\in\{0,1\}$ and let $\...
- 17.6k
13 votes
2 answers
725 views
$q$ as a prime power and a root of unity
The number of points on the $(n-1)$-dimensional projective space $P^{n-1}(\mathbb{F}_q)$ over a finite field $\mathbb{F}_q$ is the $q$-integer $$[n]_q := \frac{q^n-1}{q-1}.$$ In analogy, the number of ...
- 1,440
3 votes
1 answer
432 views
Roots of anti-palindromic polynomial if coefficients are odd.
This is in continuation of the question asked in this earlier post here. Given an anti palindromic polynomial of degree $n$ with odd coefficients, does it have roots on the unit circle?
- 541
3 votes
0 answers
181 views
Modular root of $-1$
Take two distinct coprime integers $a,b$ and take $q=a^2+b^2$. Consider the equation $(xy^{-1})^2\equiv-1\bmod q$. The solutions are two roots which correspond to $(x,y)=(a,b)$ and $(x,y)=(b,a)$. Look ...
- 1
4 votes
0 answers
345 views
power series and roots of unity
Let $p$ be an odd prime and $X$ and $Y$ be subsets of $p^{th}$ roots of unity, $|X|=|Y|=n,X\neq Y.$ Let $f(t)=\sum_{x\in X}x^{t}-\sum_{y\in Y}y^{t}$. If $f(t)=at^k+o(t^k)$ is the power series ...
- 205
5 votes
0 answers
137 views
Sign preserving Galois automorphisms
I have an algebraic number $\alpha \in \mathbb{Q}(\zeta)$, where $\zeta^n = 1$ is a root of unity (not primitive) given as a linear combination of powers of $\zeta$, i.e, $\alpha = \sum_{i=1}^k a_i \...
- 305
6 votes
0 answers
310 views
Infinitude of cyclotomic polynomials with a certain number of terms
Let $\Phi_n$ be the $n$th cyclotomic polynomial: $${\Phi _{n}(x)=\!\!\prod _{\substack {1\leq k\leq n \\ \gcd(k,n)=1}} \!\!\big(x-e^{2i\pi {k/n}}\big).}$$ Here is a list of the first 30 cyclotomic ...
- 637
4 votes
2 answers
927 views
Summation formulas involving roots of unity to various powers
I want to know properties of the following sum: $$\sum_{j=0}^{p-1} \omega^{\beta j^2}= ~? $$ where $p$ is a prime, and $\omega^p=1$, is a $p$th root of unity (and $\beta$ is an integer between $0$ and ...
- 253
1 vote
0 answers
144 views
How to evaluate this sum of roots of unity with condition to zero
In evaluating the sum: $$\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
6 votes
1 answer
800 views
Q-binomials at roots of unity
As the title says, given a general $q$-binomial $\binom{n}{k}_q$, is there some general result regarding its value at a root of unity, $q = \exp(2\pi i r/N)$?
- 16.1k
0 votes
0 answers
240 views
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
2 votes
1 answer
222 views
Simplification of a sum with roots of unity
Let $p$ be an odd prime, $\zeta $ a primitive $p-$th root of unity and $${a_n}(x) = \sum\limits_{k = 1}^{p - 1} {\prod\limits_{j = 1}^n {\left( {1 + {\zeta ^{jk}}x} \right)} } .$$ It seems that for $...
- 5,832
4 votes
0 answers
190 views
An Optimization Problem for Exponential Polynomials
Let $\omega$ be a primitive complex $n^{th}$ root of unity. I am interested in the following quantity $$ \max_{f(n)\leq \ell \leq g(n)} \quad \min_{0<k\leq n-1} \left| 1+\omega^k+\omega^{2k}+\...
- 10.6k
2 votes
1 answer
238 views
How to prove an approximation of a combinatorics identity
How to prove that $$\sum_{k\ge 0} \binom{n}{rk} =\frac{1}{r}\sum_{j=0}^{r-1}(1+w^j)^n$$ can be approximated as $\frac{2^n}{r}$, where $n\ge 0$, $r\ge 0$, $n>r$ and $w^r=1$ is a primitive $r$-th ...
- 167
4 votes
0 answers
319 views
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 ...
- 10.6k
1 vote
1 answer
859 views
Trace 0 and Norm 1 elements in finite fields
Let $\mathbb{F}_{q^\ell}/\mathbb{F}_{q}$ be the extension of finite filed $\mathbb{F}_{q}$, where $\ell$ be a odd prime and $(\neq q)$. Take $\zeta\in\mathbb{F}_{q^\ell}$. Does there exist different $...
- 255
3 votes
1 answer
235 views
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