This tag is used for both basic and advanced questions on polynomials in any number of variables. Including, but not limited to: solving for roots, factoring, checking for irreducibility. This tag is rarely used as the only tag for a question.

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28
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0answers
977 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 ...
25
votes
0answers
900 views

On the problem of polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$

The question titled "Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$" which was posed on MathOverflow attracted quite a lot of attention (and may be the question with most wrong ...
22
votes
0answers
250 views
+50

Prove $|P(0)|\leq 2n+1$

Let $P(x)$ be a polynomial with degree $\leq n$ and $|P(x)|\leq\frac{1}{\sqrt{x}}$ for $x\in(0,1]$. Prove that $|P(0)|\leq 2n+1$. The idea should be that if $|P(0)|$ is too large, then the polynomial ...
16
votes
0answers
271 views

Irreducibility of $~\frac{x^{6k+2}-x+1}{x^2-x+1}~$ over $\mathbb Q[x]$

The Artin—Schreier polynomial $~x^n-x+1~$ is always irreducible over $\mathbb Q[x]$, unless $n=6k+2$, in which case it seems to have only two factors, one of which is always $x^2-x+1$. The ...
14
votes
0answers
692 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 ...
11
votes
0answers
89 views

Define $f(x),g(x)\in \mathbb{R}$. Prove $f(x)=g(x)$

Problem: Define $f(x),g(x)\in \mathbb{R}$ are polynomials And both of them have at least one real root and satisfy: $$f(1+x+g(x)^{2})=g(1+x+f(x)^{2})$$ Prove $f(x)=g(x)$. Rather naturally, I ...
11
votes
0answers
205 views

When does a polynomial fixing a subring imply its coefficients are in that subring?

Let $S$ be a subring of $R$. If $p$ is a polynomial with coefficients in $S$, then $p$ fixes $S$ (as a function, that is, $p(s)\in S$ for all $s\in S$). A converse statement is: If $p$ is a ...
11
votes
0answers
344 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 ...
10
votes
0answers
68 views

Property of a polynomial with no positive real roots

The following is an exercise (Exercise #3 (a), Chapter 3, page 28) from Richard Stanley's Algebraic Combinatorics. Let $P(x)$ be a nonzero polynomial with real coefficients. Show that the ...
10
votes
0answers
188 views

On the maximum number of polynomials in a certain subspace

I've already asked this question on mathoverflow, but no one answered. So I put this problem also here. Sorry for that. Let $\mathbb F_q$ be a finite field and let $e, k$ be positive integers with ...
10
votes
0answers
143 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 ...
9
votes
0answers
825 views

How to prove this polynomial inequality?

How can we prove the following? If $\frac{dP_{n}}{dz}|_{z=z_{0}}=0$ then $|P_{n}(z_{0})|<2$ for all $n>1$, where $P_{n}(z)\equiv P_{n-1}^{2}+z$ and $P_{1}\equiv z$ $z$ is in the complex plane. ...
8
votes
0answers
292 views

How to simplify this combinatorial expression?

Find \begin{eqnarray} \sum_{j\in\mathbb{N}}(n-2j)^k\binom{n}{2j-m} \end{eqnarray} Note that this question is a generalization of this one. I tried to imitate the steps in the answer given in that ...
8
votes
0answers
170 views

Bounds on derivative of real positive coefficient polynomial satisfying certain properties

While thinking about this question of Clin, I wanted to consider the polynomial: $P(z) = 1+x_1z+x_2z^2+\cdots+x_nz^n$, satisfying: (I) $1\geq x_{1}\geq x_2\geq\cdots\geq x_{n}\geq0$ and ...
8
votes
0answers
267 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 ...
7
votes
0answers
66 views

$\mathbb C[X_1, \ldots, X_n]$ is a free module over $\mathbb C[X_1, \ldots, X_n]^G$

Let $G$ be finite subgroup of $GL_n( \mathbb C )$. Let $\mathbb C[X_1, \ldots, X_n]^G$ be the set of all G-invariant polynomials of $\mathbb C[X_1, \ldots, X_n]$. Is there any rule by which we can ...
7
votes
0answers
96 views

Does it follow that any element of $J(A)$ is necessarily nilpotent?

Let $A[x]$ be the algebra of polynomials with coefficients in a $k$-algebra $A$. Assume that, for any simple $A[x]$-module $M$, one has $\text{End}_{A[x]}M = k$. Does it follow that any element of ...
7
votes
0answers
248 views

How to solve the general sextic equation with Kampé de Fériet functions?

It is frequently stated, for example on Wolfram Mathworld, that the general sextic equation $$x^6 + a_5 x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x^1 + a_0 = 0$$ can be solved in terms of Kampé de ...
7
votes
0answers
75 views

Does the inverse of a polynomial matrix have polynomial growth?

Let $M : \mathbb{R}^n \to \mathbb{R}^{n \times n}$ be a matrix-valued function whose entries $m_{ij}(x_1, \dots, x_n)$ are all multivariate polynomials with real coefficients. Suppose that ...
7
votes
0answers
344 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 ...
6
votes
0answers
55 views

Roots of iterations of polynomials

Let $f \in \Bbb Q[X]$ a polynomial, and let denote by $f^n$ the composition $\underbrace{f \circ \cdots \circ f}_{n \text{ times }}$. Let $R(f^n) \subset \Bbb C$ the roots of $f^n$. I'm interested in ...
6
votes
0answers
136 views

The cubic equation $x^3 - 4 x^2 + x + 1 =0$

The cubic polynomial $P(x) =x^3 - 4 x^2 + x + 1$ has discriminant $\Delta = 169 = 13^2$ which tells us that the extension $\mathbb{Q}(a)/\mathbb{Q}$ is normal, where $a$ is any root of the equation ...
6
votes
0answers
56 views

Can we express the roots of all polynomials in terms of roots of some special polynomials?

We can describe the roots of quadratic equations in terms of addition, subtraction, multiplication, division, and the square-root function $\sqrt a$ which computes a root of the special polynomial ...
6
votes
0answers
68 views

Is there a polynomial $p$ such that it is bijective and $ p: \mathbb{Q}^n \rightarrow \mathbb{Q}$ for $ n>1$ ??

Let us define a polynomial $p$ defined as follow $$p: \mathbb{Q}^n \rightarrow \mathbb{Q}.$$ I acrossed this question in mind. Is there a polynomial $p$ such that it is bijective and $p: ...
6
votes
0answers
191 views

Reducing multivariate rational fractions to lowest terms

I wish to simplify multivariate rational fractions to a canonical form. Thanks to some very helpful mathematically inclined people who verified that my understanding of Wikipedia was correct, I'm now ...
6
votes
0answers
117 views

Isomorphism between finite fields adjoining a root

Let $p(x)=x^3+x^2+1$ and $q(x)=x^3+x+1$ be polynomials over the field $\mathbb{Z}_2$. Let $\alpha$ be a root of $p(x)$ and $\beta$ be a root of $q(x)$. Now let $K=\mathbb{Z}_2(\alpha)$ and ...
6
votes
0answers
194 views

Elementary proof of irreducibility criterion

From ``Problems from the Book'' by Andreescu and Dospinescu, the following irreducibility criterion is presented: Let $f$ be a monic polynomial with integer coefficients and let $p$ be a prime. If ...
6
votes
0answers
143 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
157 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 $n$. ...
6
votes
0answers
441 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 ...
6
votes
0answers
127 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
150 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, ...
6
votes
0answers
688 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 ...
6
votes
0answers
142 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
109 views

The smallest non-zero integer $c$ such that $\sum\limits_{n=1}^m\pm(x+n)^6 = c$?

We have the neat equalities, I. Group 1 For $k=2,3,4,5,\dots$ $$\sum_{n=1}^{2^k}\epsilon_n(x+n)^k = 2^{\frac{k(k-1)}{2}}k! = 4,\;48,\;1536,\;\color{brown}{122880},\dots$$ for appropriate ...
5
votes
0answers
58 views

Does this simple problem using Vieta's formulas have deeper connections to elliptic curves?

A friend posed the following question to me: Suppose $p(x)=x^3+ax+b$ has one real root, $x_1$, and two non-real roots, $x_2$ and $x_3$. Compute $x_1$ in terms of $x_2$. By Vieta's formulas, ...
5
votes
0answers
41 views

A question about cyclotomic polynomials.

Let $F$ be the smallest subfield of $\mathbb{C}$ which contains $i$ and which, for every positive integer $n$, contains the unique non-negative $n$th root of every non-negative real number that it ...
5
votes
0answers
314 views

A problem with sign of coefficients of a polynomial expression

Let $f$ be a real coefficient homogeneous polynomial in $n$ undeterminates, such that $f(x_1,\cdots,x_n)>0$ whenever $x_1,...,x_n$ are non-negative real numbers, not all $0$. Then how to show ...
5
votes
0answers
126 views

A question in my mind

Suppose $\displaystyle P\in \mathbb{R}[x]$ such that : $\displaystyle P(x)=2^n$ has at least one rational root for each $n\in \mathbb{N}$. Does it follow $P$ is linear? If it does or doesn't give ...
5
votes
0answers
80 views

How does the polynomial transformation $P(x) \mapsto P(x) + c$ alter the roots of that polynomial? Specifics inside.

Consider a real quadratic polynomial $Q_k(x) = (x-\nu)(x-\omega_k) - g_k^2$. I can interpret $Q_k(x)$ as a translation of the polynomial $$ (x-\nu)(x-\omega_k) = ...
5
votes
0answers
144 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
194 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 ...
5
votes
0answers
178 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
80 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
27 views

Approximation of a rational number with values of polynomial

Given two positive integers $p,\space q$, prove that there exist a polynomial $P(x)\in\mathbb{Z}[x]$ and interval $I\subseteq\mathbb{R}$ of length $\frac{1}{q}$ such that $\forall x\in ...
4
votes
0answers
22 views

Invariants of finite groups

Let $G$ be a finite group acting linearly on $\mathbb{C}^n$ and $\mathbb{C}[X]^G$ be the ring of invariant polynomials. If $G$ is a group generated by reflections, this ring is generated by $n$ ...
4
votes
0answers
60 views

$k[x_1, \dots, x_n]$ is free iff $\mathbb{C}[x_1, \dots, x_n]^G \otimes \text{Harm}(\mathbb{R}^n, G) \to k[x_1, \dots, x_n]$ isomorphism?

For any subgroup $G \subset \text{GL}_n(\mathbb{R})$ the set $\mathbb{C}[x_1, \dots, x_n]^G$, of $G$-invariant polynomials, is a graded subalgebra of $\mathbb{C}[x_1, \dots, x_n]$, resp. the set ...
4
votes
0answers
193 views

Is determinant of a representation polynomial?

Let $V$ be a vector space over $F$, and $\pi: V \to End (V)$ a linear mapping. I want to show that the determinant of $\pi(x)$ is a polynomial on $V$. My attempt It is obvious that the determinant ...
4
votes
0answers
33 views

Is there a way to approximate a polynomial as another, binary-coefficient polynomial?

Let's say I have a polynomial: $$p(x) = \sum_{n=0}^N a_n x^n$$ where $x \in \mathbb C$. Does there exist theory and/or methods on approximating $p$ as: $$p(x) \approx \hat p(x) = \sum_{m=0}^M b_n ...
4
votes
0answers
57 views

Roots of a polynomial that is composed n times with itself

Let $f(x)=x(4x^2-3)(64x^6-96x^4+36x^2-3)$ and $f^{(n)}=f(f(f(\cdots f(x))\cdots)$ (composed with itself $n$ times). Prove that for all positive integers $n$, $f^{(n)}(x)=x$ has $9^n$ distinct ...