Questions tagged [roots-of-unity]

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107 votes
6 answers
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How small can a sum of a few roots of unity be?

Let $n$ be a large natural number, and let $z_1, \ldots, z_{10}$ be (say) ten $n^{th}$ roots of unity: $z_1^n = \ldots = z_{10}^n = 1$. Suppose that the sum $S = z_1+\ldots+z_{10}$ is non-zero. How ...
Terry Tao's user avatar
  • 108k
26 votes
0 answers
899 views

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 ...
Louis Deaett's user avatar
  • 1,513
24 votes
3 answers
3k views

Products of primitive roots of the unity

Let $m>2$ be an integer and $k=\varphi(m)$ be the number of $m$-th primitive roots of the unity. Let $\Phi = \{ \xi_1, \ldots, \xi_{k/2}\} $ be a set of $k/2$ pairwise distinct primitive $m$-th ...
Jorge Vitório Pereira's user avatar
19 votes
4 answers
3k views

An algebraic number is not a root of unity?

This problem is related to my study of the Burau representation of the braid group $B_3$: I was trying to show that certain "congruence subgroups" are of infinite index. There is an approach that ...
Alex Degtyarev's user avatar
16 votes
2 answers
1k views

Probability a polynomial has a root which is a root of unity

Consider a degree $n$ polynomial $P(x)$ with coefficients $c_i \in \{-1,0,1\}$ chosen uniformly and independently. What is the probability that $P(x)$ has a root which is a root of unity? ...
Simd's user avatar
  • 3,195
16 votes
1 answer
1k views

Evaluating a remarkable term for primes p = 5 (mod. 8)

Let $p > 3$ be a prime number, and $\zeta$ be a primitive $p$-th root of unity. I am interested in knowing the exact value of $$w_p = \prod_{a \in (\mathbb F_p^{\times})^2}(1 + \zeta^a) + \prod_{b ...
Jens Reinhold's user avatar
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 $$\sum_{k=0}^{N-1}z^{2k^2+k}$$ ...
Denis Serre's user avatar
  • 51.5k
13 votes
2 answers
618 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 ...
Henry's user avatar
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10 votes
1 answer
615 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 ...
Ira Gessel's user avatar
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10 votes
1 answer
531 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 ...
Zhi-Wei Sun's user avatar
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9 votes
1 answer
369 views

A curious Gauss-Sum type identity

Let $q=e^{2\pi i/m}$, $a\in\mathbb{R}$ and $1\leq j\leq m-1$. I would like to prove that: $$(a-1)\sum_{n=0}^{m-1} q^n\frac{\prod_{k=0}^{j-2} (q^{n+k+1}-a)}{\prod_{k=0}^{j} (aq^{n+k}-1)}=0.$$ For $j=...
Eric Rowell's user avatar
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8 votes
2 answers
979 views

Power sums of p-th roots of unity

The following question was asked by a colleague of mine. For any prime $p$ consider $$ M_p:=\min_{z_1,\dots,z_p}\max_{j,k}\left|z_1^k+\dots+z_j^k\right|,$$ where $z_1,\dots,z_p$ are the complex $p$-th ...
GH from MO's user avatar
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8 votes
0 answers
345 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 <...
Alin Tomescu's user avatar
7 votes
1 answer
2k views

Uniqueness of sums of roots of unity

Let $\zeta:=e^{\frac{2\pi i}{n}}$, with $n\geq4$, and let $2\leq k\leq n-2$. Let us suppose that the prime factorization of $n$ is $n=p_1^{\alpha_1}\cdot\dots\cdot p_s^{\alpha_s}$, with $\alpha_i>...
itotcg's user avatar
  • 73
7 votes
0 answers
147 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 ...
Ian Montague's user avatar
6 votes
3 answers
4k views

Can the sum of two roots of unity be a root of unity?

Let $p$ be prime, and $z_0, z_1, ..., z_{p-1}$ be all the $p$-th roots of unity, i.e. solutions of the equation $z^p = 1.$ Is it true or false that a combination of two (or more, in general) of the ...
Liss's user avatar
  • 145
6 votes
2 answers
469 views

Why are most coefficients of these minimal polynomials divisible by $p$?

For an odd prime $p$, let $\zeta:=e^{\frac{\pi i}p}$ and choose odd $1<n<p$. Further let $q(x)$ and $r(x)$ be integer polynomials such that $r(x)$ has no common factor with $x^n+1$, and $\xi$ ...
Wolfgang's user avatar
  • 13.2k
6 votes
1 answer
1k views

Sum of products of p-th powers of roots of 1 and monomial symmetric functions

Hello mathematicians, i'm looking for explicit computations of expressions like $$ \sum_{\substack{0\leq i,j,k<n\\i\neq j\neq k \neq i}}\zeta_n^{ip^{k_1}+jp^{k_2}+kp^{k_3}} $$ and its ...
Maurizio Monge's user avatar
6 votes
2 answers
494 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 $\...
Zhi-Wei Sun's user avatar
  • 14.4k
6 votes
1 answer
629 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, $q = \exp(2\pi i r/N)$?
Per Alexandersson's user avatar
6 votes
0 answers
124 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{...
LFM's user avatar
  • 171
6 votes
0 answers
217 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 ...
Julian Barathieu's user avatar
5 votes
1 answer
930 views

Minimal polynomial of sums of roots of unity with constant term $\pm1$

Let prime $p$ and given $\zeta_p = e^{2\pi i/p}$. It is well-known that the minimal polynomial of $x = \zeta_p + \zeta_p^{p-1}$ has a constant term either $\pm 1$ and, for certain $p$, the sum of ...
Tito Piezas III's user avatar
5 votes
1 answer
301 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}...
Zhi-Wei Sun's user avatar
  • 14.4k
5 votes
0 answers
272 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}...
LFM's user avatar
  • 171
5 votes
0 answers
119 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 \...
mathstudent42's user avatar
4 votes
2 answers
700 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 ...
guest17's user avatar
  • 253
4 votes
1 answer
606 views

Primes $p=x^2+27y^2$ and Ramanujan's $x_1^{1/3} + x_2^{1/3} + x_3^{1/3}$

I was trying to generalize, $$\sqrt[3]{\sum_{k=1}^5\cos\big(\tfrac{2^k\cdot\,2\pi}{31}\big)}+\sqrt[3]{\sum_{k=1}^5\cos\big(\tfrac{2^k\cdot\,6\pi}{31}\big)}+\sqrt[3]{\sum_{k=1}^5\cos\big(\tfrac{2^k\...
Tito Piezas III's user avatar
4 votes
2 answers
627 views

Non-standard Gauss sums

I have the following problem. Let $p$ be some prime. What is the value of \begin{equation} \sum_{k=1}^{p-1} \left(\frac{k+1}{p}\right) \omega_p^{kl}, \end{equation} where $\left(\frac{k+1}{p}\right)$ ...
Liss's user avatar
  • 145
4 votes
1 answer
1k views

Are all sums of subsets of roots of unity unique?

For a prime $p$, let $S$ be the set of all $p$-roots of unity in the complex plane. Now, consider the sum, $W(R)$ of the members of a set $R$ which is a proper subset of $S$. I suspect that $R \ne R'$ ...
Yossi Gil's user avatar
  • 143
4 votes
1 answer
578 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 ...
Asvin's user avatar
  • 7,648
4 votes
1 answer
411 views

q-th powers and roots of polynomials

Let $p,q,r$ be integers with $r\ge2$; let $f$ be a polynomial of the form $f(X) = g((X+1)^r)$, which is not a $q$-th power. Let $\omega$ be a $p$-th root of unity. Prove or disprove that the ...
grok's user avatar
  • 2,489
4 votes
1 answer
512 views

Unit in cyclotomic field

Let $n \in \mathbb{N}$ and $\zeta$ be a primitive $n$-th root of unity. I want to know for which $n$ the element $1+2(\zeta+\zeta^{-1})$ is a unit in the ring of integers of $\mathbb{Q}[\zeta]$. Can ...
guest's user avatar
  • 41
4 votes
3 answers
294 views

counting complex roots which are root of unity times a real number

Let $p(x)$ be a monic polynomial over the integers. I want to count the number of roots which have the form $\zeta \cdot r$ where $\zeta$ is a root of unity and $r$ is a real number. To count the ...
Ofir's user avatar
  • 243
4 votes
0 answers
120 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) = ...
Borys Kuca's user avatar
4 votes
0 answers
288 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 ...
DavitS's user avatar
  • 205
4 votes
0 answers
146 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}+\...
kodlu's user avatar
  • 10.1k
4 votes
0 answers
267 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 ...
kodlu's user avatar
  • 10.1k
4 votes
0 answers
337 views

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=...
Favst's user avatar
  • 1,985
3 votes
1 answer
333 views

Diophantine equation with primitive nth root of unity

Fix an $n$th primitive root of unity $\xi$. I need to understand if we can characterize in an easy way all the solutions $k \in \mathbb{Z}$ of the equation $\left|1-\left(-\frac{\xi^k - 1}{\xi-1}\...
Richard Bonne's user avatar
3 votes
1 answer
355 views

When is possible to factor a field extension into one which adds no roots of unity, followed by one which adds only roots of unity?

The answer to whether this is possible for general fields is no. However, the counterexamples used two ingredients: 1) $\Bbb Q_p$, whose extensions $K$ containing $\Bbb Q_p(\sqrt[p^e]{u})$ might not ...
Alex's user avatar
  • 197
3 votes
1 answer
243 views

Does the Lehmer quintic parameterize certain minimal polynomials of the $p$th root of unity for infinitely many $p$?

The solvable Emma Lehmer quintic is given by, $$F(y) = y^5 + n^2y^4 - (2n^3 + 6n^2 + 10n + 10)y^3 + (n^4 + 5n^3 + 11n^2 + 15n + 5)y^2 + (n^3 + 4n^2 + 10n + 10)y + 1 = 0$$ with discriminant $D = (7 + ...
Tito Piezas III's user avatar
3 votes
1 answer
317 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 ...
Tito Piezas III's user avatar
3 votes
1 answer
316 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?
Student's user avatar
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3 votes
0 answers
185 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 ...
BKR's user avatar
  • 31
3 votes
0 answers
160 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 ...
Turbo's user avatar
  • 13.7k
3 votes
1 answer
192 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 ...
user9278's user avatar
2 votes
1 answer
487 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)...
Keqin Liu 'Kevin''s user avatar
2 votes
1 answer
182 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 $...
Johann Cigler's user avatar
2 votes
1 answer
318 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 ...
LFM's user avatar
  • 171