All Questions
32 questions
31
votes
7
answers
6k
views
English reference for a result of Kronecker?
Kronecker's paper Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten apparently proves the following result that I'd like to reference:
Let $f$ be a monic polynomial with integer ...
- 873
19
votes
3
answers
2k
views
Cyclotomic polynomials: $\Phi_n(p)$ is like $p^{\phi(n)}$ for big enough $p$, right?
Apologies in advance if this turns out to be simple. So far I haven't found a proof or a reference.
Although I like $p$ to be a prime, I can ask the following for positive integers $n$ and $p$, ...
- 3,209
19
votes
0
answers
523
views
univariate integer version of Hilbert's 17th problem
Let $f(x)$ be a polynomial of degree $d$ with integer coefficients such that $f(x)\geqslant 0$ for all real $x$. Is it necessarily true that there exists an integer $N(d)$ such that $N(d)\cdot f$ is a ...
- 109k
18
votes
1
answer
1k
views
Distinct integer roots for a degree 7+ polynomial and its derivative
Question: Is there a polynomial $f \in \mathbb{Z}[x]$ with $\deg(f) \geq 7$ such that
all roots of $f$ are distinct integers; and
all roots of $f'$ are distinct integers?
Background:
I asked a ...
- 7,831
17
votes
1
answer
3k
views
Is $x^{n}-x-1$ irreducible?
Is it true that for every $n \in \mathbb{N}$, $x^{n}-x-1$ is irreducible in $\mathbb{Z}[x]$?
The standard irreducibility criteria seem to fail.
- 11.3k
11
votes
4
answers
4k
views
Variants of Eisenstein irreducibility
In his article where he stated what we know as Eisenstein's irreducibility criterion (which actually was first proved by Schönemann, as was Scholz's reciprocity law and Hensel's Lemma), he ...
- 32.6k
10
votes
1
answer
345
views
Finding $q(x)$ such that $p(q(x))$ is reducible over $\mathbb{Q}[x]$
Let $p(x) \in \mathbb{Z}[x]$, such that $\deg (p) \ge 3$.
Can we always find $q(x) \in \mathbb{Z}[x]$, such that $\deg (q) < \deg(p)$ and $p(q(x))$ is reducible over $\mathbb{Q}[x]$?
Is there ...
- 3,153
8
votes
2
answers
354
views
Let $f \in \mathbb{Z}[x]$. Does $\bar{f}$ have as many roots in $\mathbb{F}_p$ as $f$ has in $\mathbb{C}$ for infinitely many primes $p$?
Let $f \in \mathbb{Z}[x]$ be a nonconstant polynomial. Consider $\bar{f} \in \mathbb{F}_p[x].$ Let $\rho_p$ be the number of distinct roots of $\bar{f}$ in $\mathbb{F}_p$, and let $\rho$ be the number ...
7
votes
2
answers
521
views
How large (small) can be the measure of a set where a polynomial takes small values ?
A $n$-th degree polynomial has precisely $n$ roots. So it is natural to ask the question
how large ( and small) can be the measure of a set where a polynomial takes small values ?
This, and other ...
- 1,795
7
votes
0
answers
427
views
Is there a name for these kinds of polynomials?
I've come across the following polynomials in my research and I am wondering if they have a name or if there is very much known about them:
\begin{equation}
F_{\chi}(T) = \sum_{a = 1}^{n-1} \chi(a)T^a
...
- 707
6
votes
0
answers
332
views
Criteria for irreducibility using the location of complex roots
I would like to see criteria for the irreducibility of a polynomial over $\mathbb{Z}$ based (mainly) on the location of the roots of the polynomial in the complex plane. An example of such a criterion ...
- 11.3k
5
votes
2
answers
366
views
Existence of algebraic integers with certain properties
Is the following statement true?
($\star$) Given integers $n > k > 0$, there exists a monic polynomial of degree $n$ with integer coefficients and constant term $\pm 1$, irreducible over $\...
- 2,479
5
votes
1
answer
351
views
Divisibility of certain polynomials
Consider the finite sums
$$F_n(q)=\sum_{k=1}^nq^{\binom{k}2}$$
with exponents the triangular numbers $\binom{k}2$. When $n$ is odd, it appears that $F_n(q)$ does not factorize over $\mathbb{Z}[q]$. On ...
- 43.2k
5
votes
0
answers
775
views
A conjecture about the degrees of special polynomials
Define the congruence "modulo m" on exponential Taylor series as
$$
\sum_{n=0}^\infty \frac{a_n}{n!}x^n \equiv \sum_{n=0}^\infty \frac{b_n}{n!} x^n \mod m \iff \forall n: \frac{a_n-b_n}{m}\in \mathbb{...
- 237
4
votes
1
answer
539
views
A (mild?) question on the number of monomials
Let $[n]_q=\frac{1-q^n}{1-q}$ with $[0]_q=0$. Recall the $q$-factorials $[n]_q!=[1]_q[2]_q\cdots[n]_q$ (with $[0]_q!=1$) and the $q$-binomials
$$\binom{n}k_q=\frac{[n]_q!}{[k]_q!\,[n-k]_q!}.$$
Now, ...
- 43.2k
4
votes
1
answer
325
views
Irreducible monic polynomials
I am looking for criteria for the irreducibility of monic polynomials with constant term $\pm1$ over $\mathbb Q$. Eisenstein's criterion clearly doesn't apply here.
For instance, for the family of ...
- 2,135
4
votes
1
answer
271
views
The highest power of $2$ dividing a polynomial evaluated at $x=3$
Let $\nu_2(a)$ be the $2$-adic valuation of an integer $x$, i.e. the largest power $t$ such that $2^t$ divides $x$.
Define the operator $D=x\frac{d}{dx}$ and the polynomial $\Phi_k(x)=\frac{x^{k+1}-1}{...
- 43.2k
4
votes
0
answers
186
views
A problem in the spirit of P. Borwein's polynomials
A well-known conjecture (now a theorem) of P. Borwein (see Wang and Krattenthaler - An asymptotic approach to Borwein-type sign pattern theorems) states:
For all positive integers $n$, the sign ...
- 43.2k
4
votes
0
answers
500
views
Zeros of polynomials modulo a non-prime
Suppose I have a set $S$ and I want to find a polynomial $p$ such that $p(s) = 0 \mod n$ if $s \in S$, and that it is non-zero modulo $n$ otherwise.
In the literature such an $S$ is sometimes called ...
3
votes
2
answers
459
views
Short sequence beats long sequence
I have encountered some comparison between two binomial sums. It was amusing how the one with "fewer" summands exceeds (in value) than the other which consists of many more terms. In fact, ...
- 43.2k
3
votes
1
answer
447
views
A number array related to colored necklaces and the primes
I stumbled upon entry OEIS-A208535 on the enumeration of certain kinds of colored necklaces and noticed that the integers for the odd prime rows of the table there seem to be given by the Moreau ...
- 10.5k
3
votes
1
answer
194
views
Divergence of a series related to Schinzel's hypothesis H
The Series
Consider the series identity
$$\Phi(s) = \sum_{n=1}^\infty \frac{\mu(n) (\log n)^k}{n^s} \sum_{r \in R(n)} \zeta(s,r/n) = \sum_{n=1}^\infty \frac{\Lambda_k'(f(n))}{n^s}$$
$$R(n) = \left\...
- 33
3
votes
0
answers
115
views
p-adic density of the image of a polynomial
Let $P(x)$ be a polynomial with integer coefficients, and let $p$ be a prime number. Recently, a user of MO proved that the limit
$$\delta_p(P) := \lim_{n \to \infty} \frac{|\{P(x) \bmod p^n : x = 1,\...
- 453
3
votes
0
answers
119
views
Furtwängler's family of irreducible polynomials
In the question Examples of nice families of irreducible polynomials over Z, user trew mentions a family of irreducible polynomials over the integers of the following form:
$$ p(x) = x^4 \prod_{i=1}^{...
- 125
3
votes
0
answers
195
views
Congruence for the polynomials $(t+1)^n$
An interesting polynomial congruence is given by
$$A_n(t^m)\equiv \left(\frac{1+t+\cdots+t^{m-1}}m\right)^{n+1}A_n(t) \qquad \mod (t-1)^{n+1}, \tag1$$
where $A_n(t)$ are the Eulerian polynomials with ...
- 43.2k
3
votes
0
answers
408
views
The second conjecture about the degrees of special polynomials
Define the congruence "modulo m" on exponential Taylor series following the previous post (A conjecture about the degrees of special polynomials)
It has been conjectured, that if we define the ...
- 237
2
votes
1
answer
192
views
A Vandermonde-type system
For a prime $p$ and $a_1,\dotsc,a_n\in\mathbb F_p^\times$, consider the system of equations
$$ \begin{cases}
\begin{align}
a_1 + \dotsb + a_n &= 0 \\
a_1x_1 + \dotsb + a_nx_n &...
- 23k
1
vote
0
answers
45
views
Proper Pisot n-tuples
Recall that x is a Pisot number if it is real and x>1, while all of its conjugates have magnitude less than 1. Then $\{(x)^k\}$ (where $\{\cdot\}$ is the fractional part of x) approaches 0 ...
- 680
1
vote
0
answers
159
views
A follow up on Bergeron's conjecture and a question
We say two polynomials satisfy $P(x)\geq Q(x)$ iff $P(x)-Q(x)$ has non-negative coefficients. Recall $(n)_q!=\prod_{j=1}^n(1-q^j)$ and the Gaussian polynomials $\binom{n}k_q=\frac{(n)_q!}{(k)_q!(n-k)...
- 43.2k
0
votes
0
answers
86
views
Polynomials of integer coefficients that evaluated at Mersenne or Fermat numbers produce square-free integers
Mersenne numbers $M_n=2^n-1$ and Fermat numbers $F_n=2^{2^n}+1$ draw the attention of professional mathematicians to get prime constellations or statements related to primality tests for these ...
-1
votes
1
answer
142
views
If $p_n(a,b)$ is a rational number (or integer) for 3 consecutive values of $n$ then every $p_n(a,b)$ is
Let $a$ and $b$ be two real numbers and $p_n(x,y)$ the polynomial: $$p_n(x,y)=\sum_{i=0}^{n-1}x^{n-1-i}y^{i},$$ where $n$ is a positive integer.
In a previous post I asked if $p_n(a,b)$ was a ...
- 3,153
-4
votes
2
answers
6k
views
Factorizing polynomials of several variables (in a different perespective)
I am looking for factorization of polynomials of several variables in the way outlined below.
Consider a second degree polynomial of two variables over the complex numbers.
"P(x,y) = Ax^2 + Bxy + Cy^...