Polynomials are expressions like $15x^3 - 14x^2 + 8$. Questions tagged with this concern common operations on polynomials, like adding, multiplying, polynomial long division, factoring and solving for roots.

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645 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 ...
12
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503 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
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284 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
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154 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 ...
10
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141 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 ...
9
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86 views

Is There An Injective Cubic Polynomial $\mathbb Z^2 \rightarrow \mathbb Z$?

Earlier, I was curious about whether a polynomial mapping $\mathbb Z^2\rightarrow\mathbb Z$ could be injective, and if so, what the minimum degree of such a polynomial could be. I've managed to ...
9
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118 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
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124 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 ...
8
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180 views

Conjecture: Tract version of Gauss--Lucas Theorem for higher derivatives.

The Gauss--Lucas Theorem states that all zeros of a degree $n$ complex polynomial $p(z)$ are contained in the convex hull of the zeros of $p$. By iteration, this implies that the zeros of ...
8
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74 views

Minimising an expression - involving polynomial

I found this one on a forum but it has been unanswered from long there. I am curious to know if there is a solution to this problem. Here it is: Let n be a positive integer. Determine the smallest ...
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250 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
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46 views

A polynomial that annihilates two other

While studying, I found the following problem: Let $f, g \in F[t]$. Prove that $\exists p \in F[x, y], p \neq 0 : p(f(t), g(t)) = 0$ I'd thank any hints that point me in the right direction.
6
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68 views

If all convex combinations of $p(x)$ and $q(x)$ have real roots, then $p,q$ have a common interlacing poly

I heard this result in a talk the other day: Suppose $p$ and $q$ are polynomials. Suppose $p$ is a polynomial of degree $n$ and $q$ a polynomial of degree $n-1$. Call $q$ and interlacer of $p$ if the ...
6
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164 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
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128 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 ...
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118 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
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145 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
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123 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
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121 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
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34 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 ...
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35 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
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304 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
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122 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
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166 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 ...
5
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59 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
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121 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 $n$. ...
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292 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
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124 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 ...
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162 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
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558 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 ...
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75 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 + ...
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51 views

On the location of the roots of a polynomial

Consider the following two polynomials \begin{align} p(s)&:=s^n+\alpha_{n-1}s^{n-1}+\cdots+\alpha_1s+\alpha_0,\\ q(s)&:=s^{n-1}+\alpha_{n-1}s^{n-2}+\cdots+\alpha_2 s+\alpha_1, \end{align} ...
4
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24 views

How to find out if a polynomial equation has real solutions?

I have a polynomial equation of $N$th order. The coefficients of the equation are parametrized by two variables, let's call them $a$ and $b$, both of which are real and positive. For general $N$, I ...
4
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101 views

$15a+6b+4c+8d=0$ implies $ax^3+bx^2+cx+d$ has a positive root

Let $a,b,c,d$ be real numbers such that $15a+6b+4c+8d=0$. Show that $f(x)=ax^3+bx^2+cx+d$ has a positive root. I want to try to use the intermediate value theorem, showing that $f(s)<0$ and ...
4
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66 views

Writing the roots of a polynomial with varying coefficients as continuous functions?

Consider the monic polynomial $$p_{\zeta}(z) = z^n + a_{n-1}(\zeta)z^{n-1} + \dots + a_0(\zeta), $$ where the $a_{i}$'s are continuous functions defined over $\mathbb{C}$. As is well known, the ...
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41 views

Set of Metapolynomials is closed under multiplication

We say that a function $f:\mathbb{R}^k \rightarrow \mathbb{R}$ is a metapolynomial if, for some positive integer $m$ and $n$, it can be represented in the form $$f(x_1,\cdots , x_k ...
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74 views

Irreducibility of some polynomial

Let $p(x) = (1+ \cdots +x^k)^2 + (1+ \cdots +x^k) + 1$, for some $k \geq 2$ fixed. I would like to know if $p(x)$ is irreducible in $\mathbb{Q}[x]$.
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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 ...
4
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99 views

All roots of a polynomial lie on a circle.

I'm stuck in the following problem and I need your help to solve it. Given a number $\alpha$, $0 < \alpha < 1$. $A_j(x)$ is a sequence of polynomials of $x^{-1}$ such that: $A_0(x) = 1; \\ ...
4
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57 views

The ring of homogeneous polynomials

I think I found an error in my textbook, but I am not completely sure. The book is Hulek, Elementary algebraic geometry, pag. 73. There is a theorem showing that $U_i$ and the affine space ...
4
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124 views

norm of differential operator on $P^n[0,1]$

Consider the space $P^n[0,1]$ of polynomials of degree $\leq n$ on $[0,1]$, equipped with the sup norm. Now, this is a finite dimensional space, so all linear operators have to be continuous, hence ...
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67 views

Solve $f(x)\mid g^2(x)+1$ in $\mathbb Z[x]$

We know that if $p\in \mathbb P$ and $p\equiv 1\bmod 4$ then we can find $t\in\mathbb Z$ such that $p\mid t^2+1.$ For what polynomial $f(x)\in \mathbb Z[x]$, we can find $g(x)\in \mathbb Z[x]$ ...
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65 views

Can we say anything about the structure of the semigroup of non-coprime pairs after this?

Let $S = \{(a,b) : \ a, b \in \Bbb{Z} \wedge \gcd(a,b) \neq 1 \}$. Then it forms a semigroup under componentwise multiplication and if we add an exception, that even though $\gcd(1,1) = 1$, we ...
4
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78 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 ...
4
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49 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
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110 views

When is an intersection of varieties finite

Consider the general Bezout's theorem: If $p_1 \ldots p_n$ are polynomials with degrees $d_1,\ldots, d_n$ in $\mathbb{R}[x_1,\ldots,x_n]$, with $V = \{a=(a_1,\ldots, a_n) | p_i(a) = 0, \forall i\}$ ...
4
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162 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
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86 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
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141 views

Multivariable polynomial decomposition

There is a beautiful theorem that allows us to decompose a polynomial of several variables into the sum of polynomials of a single variable. Let $P_{N}(x)$ be a multivarible polynomial of degree ...
4
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60 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 begins shows that, for each $k\in \mathbb{N}$, there is a unique polynomial $P_k(x)$ of degree $k$ whose coefficients are in $\mathbb{Q}(q)$, the field of rational functions, ...