Polynomials are expressions like $15x^3 - 14x^2 + 8$. Questions tagged with this concern common operations on polynomials, like adding, multiplying, dividing, factoring and solving for roots.
42
votes
0answers
1k views
All polynomials with no natural roots and integer coefficients such that $\phi(n)|\phi(P(n))$
Let $P(x)$ be a polynomial with integer coefficients such that the equation $P(x)=0$ has no positive integer solutions. Find all polynomials $P(x)$ such that for all positive integers $n$ we have ...
15
votes
0answers
109 views
Rational roots of polynomials
Can one construct a sequence $(a_k)_{k\geqslant 0}$ of rational numbers such that, for every positive integer $n$ the polynomial $a_nX^n+a_{n-1}X^{n-1}+\cdots +a_0$ have exactly $n$ distinct rational ...
13
votes
0answers
134 views
Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes
Find all polynomials $\sum_{k=0}^na_kx^k$, where $a_k=\pm2$ or $a_k=\pm1$, and $0\leq k\leq n,1\leq n<\infty$, such that they have only real zeroes.
I've been thinking about this question, but ...
10
votes
0answers
520 views
Is $ f_n=\frac{(x+1)^n-(x^n+1)}{x}$ irreducible over $\mathbf{Z}$ for arbitrary $n$?
In this document on page $3$ I found an interesting polynomial:
$$f_n=\frac{(x+1)^n-(x^n+1)}{x}$$
Question is whether this polynomial is irreducible over $\mathbf{Q}$ for arbitrary $n \geq 1$ ?
In ...
8
votes
0answers
148 views
Is $\bigl(X(X-a)(X-b)\bigr)^{2^n} +1$ an irreducible polynomial over $\mathbb{Q}[X]$?
Let $a, b \in \mathbb{Q}$, with $a\neq b$ and $ab\neq 0$, and $n$ a positive integer.
Is the polynomial $\bigl(X(X-a)(X-b)\bigr)^{2^n} +1$ irreducible over $\mathbb{Q}[X]$?
I know that ...
8
votes
0answers
136 views
Do there exist polynomials $f,g$ such that $\mathbb{C}[a,b,c]\le\mathbb{C}[f,g]$ for $a,b,c$ given polynomials?
I want to prove something bigger than the problem in the title and I want to create a lemma that is useful for the solution of the problem. But I am unable to prove (or give a counterexample) the ...
8
votes
0answers
325 views
The radical solution of a solvable 17th degree equation
(The question is at the bottom of the post.) Here's a "natural" solvable 17-th deg eqn with small coefficients:
$$\begin{align*}
x^{17}-6 x^{16}&-24 x^{15}-42 x^{14}-31 x^{13}-23 x^{12}-7 ...
7
votes
0answers
58 views
Why do you need to introduce complex numbers in order to solve cubics by radicals?
Historically, complex numbers were first introduced (by Bombelli) in order to provide a method for solving the cubic by radicals. As far as I can tell, previous solutions of the cubic by radicals ...
7
votes
0answers
150 views
What is the dimension of this algebraic variety?
Let $\mathbb K$ be a number field of degree $n$ over $\mathbb Q$, and let
$\alpha_1,\alpha_2, \ldots ,\alpha_n$ be a $\mathbb Q$-basis of $\mathbb K$. Then there
are coefficients $(c^{ij}_k)$ (where ...
7
votes
0answers
214 views
On the continuation of a polynomial
This exrcise is from the first section of Marden:
Exercise 12. Let the interior of a piecewise regular curve $C$ contain the origin $\cal O$ and be star-shaped with respect to $\cal O$. If the ...
6
votes
0answers
98 views
Determinant vanishing over polynomial ring
Let $R=\mathbb C[t_1,\ldots,t_N]$ be a polynomial ring in some number of variables. Assume that $f_{ij}\in R$ are homogeneous linear polynomials for $1\le i,j\le n$. If $\det(f_{ij})=0$, I can ...
6
votes
0answers
121 views
Distribution of roots of complex polynomials
I generated random quadratic and cubic polynomials with coefficients in $\mathbb{C}$
uniformly distributed in the unit disk $|z| \le 1$. The distribution of the roots of 10000
of these polynomials are ...
6
votes
0answers
107 views
If $F(a_1,\ldots,a_k)=0$ whenever $a_1,\ldots,a_k$ are integers such that $f(x)=x^k-a_1x^{k-1}-\cdots-a_k$ is irreducible, then $F\equiv0$
I'm trying to understand a proof of the following theorem (from section II of Hall's paper An Isomorphism Between Linear Recurring Sequences and Algebraic Rings):
If $F(a_1, \ldots, a_k)$ is a ...
6
votes
0answers
93 views
Check my proof of the polynomial uniqueness
Problem 169 from the book I.M. Gelfand, "Algebra".
"Assume that $x_1, \ldots , x_{10}$ are different numbers, and $y_1 , \ldots , y_{10}$ are arbitrary numbers. Prove that there is one and only one ...
6
votes
0answers
79 views
Monotonic version of Weierstrass approximation theorem
Let $f\in\mathcal{C}^1([0,1])$ be an increasing function over $[0,1]$.
Prove or disprove the existence of a sequence of real polynomials $\{p_n(x)\}_{n\in\mathbb{N}}$ with the properties:
$p_n(x)$ ...
6
votes
0answers
105 views
On the order of $\mathbb{Z}[X]/(f,g)$ and the resultant $R(f,g)$.
I suspect that $\#\mathbb{Z}[X]/(f,g)=|R(f,g)|$ holds for any two non-constant polynomials $f,g\in\mathbb{Z}[X]$, where $R(f,g)$ is the resultant of $f$ and $g$. I am however unable to prove it. I'd ...
6
votes
0answers
110 views
Bounding the “complexity” of irreducible factors of an integer polynomial
Given an integer polynomial $P(x) = a_0 + a_1 x + \cdots + a_n x^n$, there ought to be a reasonable bound on the "complexity" of its possible irreducible integer polynomial factors that allows us to ...
5
votes
0answers
39 views
Fejer-Riesz Lemma
I'm trying to apply the Fejer-Riesz Lemma constructively. The lemma says that for a Laurent polynomial $a(z) = \sum_{-n}^na_jz^j$ with $a_j = \bar a_{-j}$ and $a(e^{i\theta})\geq0$ on the complex unit ...
5
votes
0answers
79 views
Generating Functions, Recursive Polynomials
At the CMFT international conference in Turkey (2009), the following open problem was given:
Show that $$p_n(x):=\sum_{k=0}^n \frac{(n-k)^k}{k!}x^{n-k}$$ has only real simple zeros for every $n$. ...
5
votes
0answers
82 views
Determine the general solution for $2\cos^2x-5\cos x+2=0$
Determine the general solution for $ 2\cos^{2}x-5\cos x+2=0$
My attempt:
$2u^2 - 5u + 2 = 0$
$(2u - 1)(u - 2) = 0$
$u = \frac{1}{2}$ or $u = 2$
$\cos(x) = \frac{1}{2}$ or $\cos(x) = 2$
The ...
5
votes
0answers
93 views
Runge's phenomen: interpolation error using Chebyshev nodes oscillates
We're trying to approximate the Runge function $f(x) = \dfrac{1}{1+25x^2}$ using Chebyshev nodes. When calculating the interpolation error, using different degrees ranging from 0 to 50, we get the ...
5
votes
0answers
105 views
Proof that $t^8+2t^6+4t^4+t^2+1$ is reducible in $\mathbb{F}_p$
Prove that the polynomial $t^8+2t^6+4t^4+t^2+1$ is reducible in $\Bbb F_p$, for all $p\in \Bbb P$.
Here are some examples:
$t^8+2t^6+4t^4+t^2+1=(1 + t + t^4)^2\pmod{2}$
$t^8+2t^6+4t^4+t^2+1=(1 + t) (2 ...
5
votes
0answers
140 views
Irreducibility of polynomials via Frobenius map
I am having trouble trying to show this:
Let $f \in \mathbb{F}_p[x]$ be a non-constant polynomial and let $F$ denote the Frobenius map $F: R \rightarrow R$ where $R = \mathbb F_p[x]/(f)$. Prove ...
5
votes
0answers
102 views
Computable Criteria to check whether a given basis is a Gröbner Basis
In an upcoming exam we have to do Gröbnber-Basis computation with Buchberger's algorithm. A typical example looks like this:
$$ \langle f_1,f_2 \rangle $$
Then I compute the S-Polynomial ...
5
votes
0answers
92 views
relationship between solution of quintic in terms of $_{4}F_{3}$ hypergeometric function and theta functions
There is one approach (Bring radical/method of differential resolvents) to the general solution to the quintic that gives the solution for a particular root $v\in\{v_{1},v_{2},v_{3},v_{4},v_{5}\}$ in ...
5
votes
0answers
72 views
Restriction of trivariate polynomial to $1$ variable
Let $p(x,y,z): \mathbb{F}^3 \to \mathbb{F}$ be a trivariate polynomial of degree $d \ll |\mathbb{F}|$. We choose uniformly at random an affine $1$-dimentional space $\ell = \{(a_1,a_2,a_3)t + ...
4
votes
0answers
35 views
Existence of Solutions of Two Cubic Equations in a Particular Region
If I have two cubic equations in two variables, $ax^3 + bx^2 y + cxy^2+\dots=0$ and another one with different coefficients, and I know that $(x,y)=(0,0)$ or $(1,1)$ are solutions, are there any nice, ...
4
votes
0answers
67 views
“Convex” polynomials
Let me define "convex" polynomials, as the smallest class $\mathcal{C}$ of functions $p:\mathbb{R}\rightarrow \mathbb{R}$ defined (inductively) as:
UPDATED (case 0 was missing):
0) $p(x)=x$, i.e., ...
4
votes
0answers
72 views
Polynomial bound
Let $P(x)=a_4 x^4+a_3 x^3+a_2 x^2+a_1 x+a_0$ such that
$$\forall i\in \{0, 1, 2, 3, 4\};\phantom{;}a_i\in\mathbb{Z} \wedge |a_i|\leq T\phantom{.}(T\in\mathbb{Z}^+ )$$
Suppose that $P(x)> 0$ for all ...
4
votes
0answers
130 views
Generalized conjugation of polynomials with coefficients in an associative algebra
Suppose $\mathcal{A}$ is an associative algebra over $\mathbb{R}$. Furthermore, let $f(x_1, \dots , x_n) \in \mathcal{A}[x_1, \dots , x_n]$.
Preliminary Question: Is it possible to find $g(x_1, ...
4
votes
0answers
392 views
Gauss-Lucas Theorem (roots of derivatives)
Gauss-Lucas Theorem states:
"Let f be a polynomial and $f'$ the derivative of $f$. Then the theorem states that the $n-1$ roots of $f'$ all lie within the convex hull of the $n$ roots ...
4
votes
0answers
54 views
Necessary and sufficient condition for $f(q^n)$ to be in $\mathbb{Z}[q,q^{-1}]$ when $f\in\mathbb{Q}(q)[x]$?
In this question, user bgins shows that for each $k$ there is a unique polynomial $P_k(x)$ of degree $k$ whose coefficients are in $\mathbb{Q}(q)$, the field of rational functions, such that ...
4
votes
0answers
64 views
About $t$-analogue of the Euler polynomials.
A certain way to define the $t$-analogue of the Euler polynomials $C_n(x)$ is by
$$
C_n(x,t)=\sum_{\pi\in S_n}x^{\text{des}(\pi)+1}t^{\text{maj}(\pi)}
$$
where $des(\pi)$ is the descents in $\pi$, ...
4
votes
0answers
166 views
A functional recursion problem..do you have any idea?
I have a problem which is related to algebra and polynomials. I would be very grateful if any of you could give a hand to solve it. Here is the problem: Consider the function $f_0(x) =x(1-x)$ and for ...
3
votes
0answers
34 views
(Integer) Variant of Hilbert’s irreducibility theorem
Let $P\in{\mathbb Q}[X,Y]$ such that $P(x,.)$ has an integer root for any integer
$x\in{\mathbb Z}$. Does it follow that $P$ has a factor of the form
$Y-Q(X)$ for some $Q\in{\mathbb Q}[X]$ such that ...
3
votes
0answers
46 views
How does Dedekind's Theorem work for a prime dividing the discriminant of a number field?
Let $f \in \mathbb{Z}[x]$ be an irreducible monic polynomial, let $N$ be its splitting field, and let $G$ be the Galois group of the extension $N/\mathbb{Q}$. Let $p$ be a prime dividing the ...
3
votes
0answers
42 views
Polynomials dense in $C^\infty(U, V)$?
Let $V$ be a Banach space and $U$ an open subset of $\mathbb{R}^n$. Let $C^\infty(U, V)$ denote the Fréchet space of $V$-valued smooth functions on $U$ with seminorms
$$ \| u\|_{\alpha,K} = \sup_{y ...
3
votes
0answers
70 views
What does $x_n = s\, x_{n-1}$ mean in the components of recurrence?
Say you have a reucrrence $x_{n+1} = 3x_n+2$. Though, it is a inhomogenous, it can be represented by a linear system
$$\begin{bmatrix} x_{n+1}\\1\end{bmatrix} = \begin{bmatrix} 3 & 2 \\ 0 & ...
3
votes
0answers
88 views
A basis of the symmetric power consisting of powers
Let $V$ be a complex vector space of dimension $n$. Denote by $v_1\odot\cdots\odot v_k$ the image of $v_1\otimes\cdots\otimes v_k$ in the symmetric power $\newcommand{\Sym}{\mathrm{Sym}}\Sym^k(V)$. It ...
3
votes
0answers
67 views
Deligne and the four Weil statements about polynomials over finite fields?
This article mentions he recently won 3 million of some money. Does anyone have a link to the original Deligne paper that proves this, and could anyone give a basic rundown of how "counting points on ...
3
votes
0answers
74 views
question on minimal polynomial
Let $\alpha$ be a root of $x^3+3x-1$ and let $\beta$ be a root of $x^3-x+2$. Find the minimal polynomial of $\alpha^2+\beta$.
My attempt to solution was this: i found a monic polynomial with integer ...
3
votes
0answers
152 views
Understanding how Prime Polynomials are applied to LFSRs?
In doing some research on LFSRs I understand that a primitive polynomial can determine what taps to be used to create an LFSR that has as many bits as the degree of the polynomial that will cycle ...
3
votes
0answers
74 views
Prove (*) by induction on k.
Challenge: For linear systems with constant coefficients, in some sense we "never need more" than the so-called exponential polynomials, meaning expressions in the form
$$\sum_{i=1}^m ...
3
votes
0answers
60 views
Given an integer, how can I detect the nearest integer perfect power efficiently?
If you give me an integer N, how can I detect the nearest integer perfect power, larger or smaller than N?
In other words, the perfect power the distance between N and which is less than the ...
3
votes
0answers
93 views
linear combinations of Chebyshev polynomials of first and second kind
I do not know if this problem was considered before.
Prove that there are infinitely many pairs $(a,b)$ of mutually prime integers such that none of the polynomials
$P_n(x) = a*T_n(x) +b*U_n(x) $
...
3
votes
0answers
59 views
Existence of roots of a polynomial equation when coefficients have varying weights
I have two $n-$degree polynomials $f_{1}(p)$ and $f_{2}(p)$, where the domain of $p\in[0,1]$. I know that $\exists$ $0 < p_{1} < 1$ such that:
$f_{1}(p_{1}) = f_{2}(p_{1})$.
Let ...
3
votes
0answers
78 views
Vague question about polynomials and symmetry
The problem to find polynomials $f(x,y)$ such that $f(x,y) - f(y,x) = 0$ can be 'solved' by characterizing the solutions as all polynomials in $xy$ and $x+y$ (according to the fundamental theorem of ...
3
votes
0answers
166 views
Closed-form expression for sum of Vandermonde matrix elements
Given the Vandermonde matrix:
$$\begin{pmatrix}1^0 & 1^1 & 1^2 & ... & 1^n \\
2^0 & 2^1 & 2^2 & ... & 2^n \\
\vdots & \vdots & \vdots & \ddots & ...
3
votes
0answers
26 views
Study a particular polynomial sequence
Let us define the following sequence of polynomials for every two non-negative integers $i,d$: $$s_i^d(w)=\sum_{j=0}^{d+1} (-1)^j {d+1\choose j} (j+1)^i w^{d+1-j}.$$
Conjecture: The sequence ...
3
votes
0answers
127 views
Positive definite completion of a matrix
Suppose we have a real, symmetric matrix $A(x_1,x_2,x_3)$ given by
\begin{pmatrix}
a_{1,1} & a_{1,2} & x_1 & x_2 \\
a_{2,1} & a_{2,2} & a_{2,3} & x_3 \\
x_1 & a_{3,2} & ...
