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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Identifying a sequence of numbers from an optimization problem in $L^1$

Question Does there exist general closed form solutions (or some sort of recurrence relation) to the system of equations: $$\begin{align} x_0 &= -1\\ x_{k+1} &= 1\\ \sum_{j = 0}^k (-1)^j (x_{...
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0answers
20 views

Linear transformation of vector ARMA processes

Can someone help me to solve the following problem. Referring to the one above the bottom equation: I was managed to get the left hand side and first term of the right hand side. But couldn't ...
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0answers
22 views

Roots of polynomial in terms of “odd radicals”

I'm trying to show that it is not possible to find a formula for the roots of $x²-S_1x+S_2$ only in terms of odd radicals in $C(x,y)$. Here $S_1$ and $S_2$ are the symmetric polynomials in terms of ...
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3answers
57 views

Ideals Generated by polynomials

So I am currently studying a course in commutative algebra and the main object that we are looking at are ideals generated by polynomials in n variables. But the one thing I don't understand when ...
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1answer
83 views

example of proper ideal of C[x,y]

I am stuck in this problem for a while, and the main idea will be important for some exercises, so I really want to know how to find an example like this I need an example of an proper ideal, let'...
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2answers
65 views

Roots of a Polynomial Minus It's Constant Term

Suppose we have a sequence of integers $a_1,\dots,a_n$. Is there any way to determine the roots of the polynomial $$P(x) = (x+a_1)\dots(x+a_n) - a_1\dots a_n$$ Clearly $P(0) = 0$, but can anything ...
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1answer
99 views

Prove that the following cubic polynomial is irreducible over $\mathbb Q$

Let $a\neq b$ $|a,b \in\mathbb N$ and let $P(x)=x^3+ax^2+bx+1$ Show that $P$ is irreducible over $\mathbb Q$. I tried writing $(x^2+dx+e)(x+f)=x^3+ax^2+bx+1$ to find a contradiction. Found ...
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0answers
45 views

Taylor expansion of a power function

I was wondering about Taylor expansions of functions of the form $x^p$, where p is a real number, about $x = 0$. It seems clear how to do it about any other point, but what happens to the series as I ...
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1answer
36 views

Integer-valued polynomial question

Let us have an $f(x)$ Integer-valued polynomial, which gains the value $1$ in $4$ different places. Prove, that in that case, it can't gain the value $-1$ on integer places. I tried with $f(x)-1=0$, ...
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2answers
101 views

Is there a link between the Bunyakovsky conjecture and the Twin Prime conjecture?

Can the proof of one conjecture be considered a proof of the other conjecture? The general method of building an infinite number of prime producing quadratic polynomials was given in the link below....
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1answer
74 views

Irreducibility in $k((t))[y]$

Let $k$ be an algebraically closed field of char $0$ and suppose $f(y) \in k[y]$ (need not be monic). Let $t$ be an indeterminate and consider the fraction field $k((t))$ of power series ring $k[[t]]$....
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2answers
62 views

Prove that $ (a+b\sqrt{2})^n $ is of the form $k+l\sqrt{2}$.(a,b,k,l,n are integers; n>1)

I have previously proved it for n=1. Using induction, assume $(a+b\sqrt{2})^{x-1}$ is true; it is of the form $k+l\sqrt{2}$. for $(a+b\sqrt{2})^x$; how do i proceed from here? Binomial theorem for ...
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2answers
70 views

Help me to prove this statement about quadratic equations? (from Gelfand's Algebra).

$ x^2+px+q=0 ${p,q are integers; a,b are roots}. Prove $a^n+b^n$(n is any natural number) is an integer. This is the third part of the problem.I have previously proved that $a^2+b^2$ and $a^3+b^3$ ...
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1answer
45 views

Find all Real polynomials.

"Find all real polynomials for which $f(2) = 3, f(3) = 5$ and $f(5) = 2$." Well my first thought was, since we have three points i can determine a polynomial of second degree such that it satisfies ...
2
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2answers
56 views

Find $P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 ,\ n\ge 1$ has $n$ roots $x_1,x_2,\ldots,x_n \le -1$ and such that $a_0^2+a_1a_n=a_n^2+a_0a_{n-1}.$

Let $P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 ,\ n\ge 1$ have $n$ roots $x_1,x_2,\ldots,x_n \le -1$ and $a_0^2+a_1a_n=a_n^2+a_0a_{n-1}$. Find all such $P(x)$ (Poland 1990). I used Viete Theorem ...
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1answer
79 views

three cubic homogeneous polynomials satisfy a cubic polynomial

Question: How can we show algebraically that three cubic homogeneous polynomials in two variables satisfy a cubic polynomial of three variables? More specifically, let $f_1(x_0,x_1),f_2(x_0,x_1),f_3(...
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1answer
103 views

how to show the derivative of the polynomial is bounded by itself in certain space.

How to prove that for every positive integer $d$, there exists $C(d)>0$, such that: For every polynomial with degree $\leq d$, we have $\max\limits_{x\in [0,1]}|p'(x)|\leq C(d)\max\limits_{x\in [0,...
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2answers
92 views

Greatest common divisor of polynomial in Finite Field(256), AES

Have assigment and will use it as example, found solution computationaly, want to understand idea. It is about SubBytes procedure in AES, particulary about finding inverse of polynomial. Suppose we ...
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1answer
89 views

Non trivial solutions of a polynomial equation

In a question a user asked for a polynomial which solves $$2P(2x^2-1)=(P(x))^2-1.$$ There are two solutions I could provide, namely the two constant ones. However in the comments to my answer it has ...
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1answer
201 views

Find a polynomial P(X)

Find a polynomial $P(x)$ such that it satisfies $$2P(2x^2-1)=(P(x))^2-1$$ How to find all of them?
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1answer
48 views

The number of zeros of a polynomial that almost changes signs

Let $p$ be a polynomial, and let $x_0, x_1, \dots, x_n$ be distinct numbers in the interval $[-1, 1]$, listed in increasing order, for which the following holds: $$ (-1)^ip(x_i) \geq 0,\hspace{1cm}i \...
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1answer
39 views

Is Gershgorin bound of roots sharp?

Gershgorin circle theorem tells that the eigenvalues of a matrix $A$ lie in the union of the associated Gershgorin circles. $A=\begin{pmatrix} 0 & 0 & \dots & 0 & -a_0 \\ 1 & 0 &...
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1answer
127 views

What does the notation $\mathbb R[x]$ mean?

What does the notation $\mathbb R[x]$ mean? I thought it was just the set $\mathbb R^n$ but then I read somewhere that my lecturer wrote $\mathbb R[x] = ${$\alpha_0 + \alpha_1x + \alpha_2x^2 + ... + \...
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1answer
85 views

Derivation of the discriminant of a cubic polynomial by algebraic manipulation.

The problem was asked before: Using Vieta's theorem for cubic equations to derive the cubic discriminant . I tried to solve it by purely algebraic manipulation but was faced with an explosion of ...
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0answers
112 views

Monic polynomial

Recently I've learned that when a given polynomial is a monic polynomial, then this polynomial root has to be a rational root. As far as I know, to figure out if a given polynomial is a monic ...
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1answer
79 views

What can we learn from prime generating polynomials?

Here's a simple polynomial that generates quite a few primes (not necessarily consecutive). $p(n) = n^2 + 23n + 23$ with $n=0,1,2... $ What can such polynomials tell us about primes? Thanks. ...
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1answer
30 views

Relation between roots an coefficients in a generic equation: $a_0+a_1\cdot x+\cdots+a_n\cdot x^n$

In a generic equation $$a_0\cdot x+a_1\cdot x^2+ a_3\cdot x^3+\cdots+a_n\cdot x^n$$ there are some relations between roots ($x_1, x_2,\ldots,x_n$) and coefficients ($a_0, a_1,\ldots,a_n$). How can i ...
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50 views

Condition Butterworth polynomial

My course states that a polynomial is a Butterworth polynomial when it satisfies the following condition: $|B(j\Omega)|=\sqrt {1+{\Omega}^{2\,n}}=\sqrt {1+{(\omega/\omega_p)}^{2\,n}}$ I'm really ...
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1answer
40 views

Is there closed form solution for this infinite polynomial or high-order polymonial?

The equation is as follows \begin{align} \sum_{N=1}^{\infty}P(N)x^N=Z, \end{align} where $P(N)$'s are real number satisfying $0\leq P(N)\leq 1$. Another equation is \begin{align} \sum_{N=1}^{\bar N}...
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1answer
50 views

why there are not polynomials $p,q$ such that $\sqrt{x^2-4}=\frac{p(x)}{q(x)}$

show that there are not polynomials $p,q$ such that $$\sqrt{x^2-4}=\dfrac{p(x)}{q(x)}$$ there a book say it is clear,because if such polynomials existed,then each zero of$x^2-4$ should have even ...
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0answers
46 views

An ideal in a ring of polynomials and a field extension.

Let $K\subseteq L$ be fields and $I$ an ideal of $K[x_1,...,x_n]$. I want to show that $IL[x_1,...,x_n]\cap K[x_1,...,x_n] =I$. The inclusion $I \subseteq IL[x_1,...,x_n]\cap K[x_1,...,x_n]$ is clear,...
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1answer
69 views

Inverse pairing function with polynomial constituents

Many bijective pairing functions $f:\mathbb N \times \mathbb N \rightarrow \mathbb N$ exists, including polynomial ones such as the Cantor pairing function $$f(n,m) = \frac{1}{2}(n + m)(n + m + 1)+m$$...
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2answers
48 views

$x^3-3x^2+(a^2+2)x-a^2$ has 3 roots $x_1,x_2,x_3$ such that $\sin \tfrac{2\pi x_1}{3}+\sin \tfrac{2\pi x_3}{3}=2\sin \tfrac{2\pi x_2}{3}$. Find $a$.

$x^3-3x^2+(a^2+2)x-a^2$ has 3 roots $x_1,x_2,x_3$ such that $\sin \dfrac{2\pi x_1}{3}+\sin \dfrac{2\pi x_3}{3}=2\sin \dfrac{2\pi x_2}{3}$. Find $a$ (Bulgari 1998)
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4answers
318 views

How can I use Fundamental Theorem of Symmetric Polynomials to factor polynomials?

How can I use The fundamental theorem of symmetric polynomials (or its proof) to factor symmetric polynomials? The link I've given to the theorem uses elaborate wordings using 'rings', 'isomorphic'...
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1answer
193 views

Proving a linear transform defined by an integral is injective

Let the fact that $I(p)(x)=\int_0^x p(s) ds$ is a linear transform from $P_4\rightarrow P_5$ be given. Prove that $I$ is injective. Would it be sufficient to just state that for any 2 polynomials,$f(...
5
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0answers
124 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 $\...
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2answers
36 views

Existence of polynomials $g$, $h$, with non-negative coefficients, such that $f(x)= \frac{g(x)}{h(x)}$ [closed]

Suppose $a$ and $b$ are real numbers such that the quadratic polynomial $f(x) = x^2 + ax + b$has no non-negative real roots. Prove that ther exist two polynomials g,h, whose coefficients are non-...
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2answers
101 views

Shamir's secret sharing interpolation problem

I try to understand this protocol - Shamir's secret sharing - threshold scheme. I got my data and I made interpolation basing on examples published on Wikipedia. You can see them below (sorry, I am ...
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1answer
155 views

Prove the extension to be a Galois Extension

Let $p$ be a prime number. $K$=$\mathbb C(x,y)$ and $F=\mathbb C(x^p,y^p)$.Then, Prove that $K/F$ is a Galois Extension. Trial: Since this $\mathbb C$ is a field of charactersitic $0$,it would be ...
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2answers
82 views

Consider $n$ numbers $a_1,…, a_n$ and $x_1,…, x_n$. Can one find a polynomial, $f(x)\in R[x]$ st $f$ path through $(x_i,a_i) $

Consider $n$ arbitrary integer numbers $a_1,\ldots, a_n$ and real numbers $x_1,\ldots, x_n$. Can one find a polynomial, $f(x)\in \mathbb{R}[x]$ such that the graph of $f$ path through $(x_1,a_1), \...
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0answers
39 views

Finite Inseparable Extension

Preparing for my Galois theory exam in may and i am faced with the following question. Give an example of a finite inseparable extension with a sketched proof of its inseparability I have the ...
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2answers
84 views

$x\in \mathbb{R} : P(x) \in \mathbb{Z} \Leftrightarrow Q(x) \in \mathbb{Z}$. Prove that $P(x)-Q(x)=c \in \mathbb{Z}$ or… [closed]

$P(x),Q(x)$ are two polynomials such that $x\in \mathbb{R} : P(x) \in \mathbb{Z} \Leftrightarrow Q(x) \in \mathbb{Z}$. Prove that $P(x)-Q(x)=c$ or $P(x)+Q(x)=d, $ where $c,d \in \mathbb{Z}$.
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5answers
186 views

Prove that equation $x^6+x^5-x^4-x^3+x^2+x-1=0$ has two real roots

Prove that equation $$x^6+x^5-x^4-x^3+x^2+x-1=0$$ has two real roots and $$x^6-x^5+x^4+x^3-x^2-x+1=0$$ has two real roots I think that: $$x^{4k+2}+x^{4k+1}-x^{4k}-x^{4k-1}+x^{4k-2}+x^{4k-3}...
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2answers
42 views

Let $ (x-1)^n\mid P(x)$ Prove that $P(x)$ has $n+1$ coefficients not zero

Let $ (x-1)^n\mid P(x)$ Prove that $P(x)$ has $n+1$ coefficients not zero It's is 1977 Bulgaria contest, i tried but not succeed
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2answers
82 views

Given some zeroes of a real polynomial of a given degree, how can one find the remaining zeroes?

Here is what the problem says: If $2$, $-\sqrt{5}$, and $3+i$ are three zeroes of a $5$th degree polynomial function with real coefficients, find the other zeroes of multiplicity $1$. I don't ...
2
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2answers
50 views

About a polynomial with complex coefficients taking integer values for sufficiently large integers

Let $f(x)$ be a polynomial with complex coefficients such that $\exists n_0 \in \mathbb Z^+$ such that $f(n) \in \mathbb Z , \forall n \ge n_0$, then is it true that $f(n) \in \mathbb Z , \forall n \...
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1answer
36 views

On the leading coefficient of a polynomial which takes integer values at every integer argument

If $f(x)$ is a polynomial with complex coefficients of degree $k$ with leading coefficient $a_k$ such that $f(n) \in \mathbb Z, \forall n \in \mathbb Z$, then is it true that $|a_k| \ge \dfrac 1{k!}$ ...
2
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1answer
51 views

relations between a set of polynomials

I have a set of polynomials. Is there a computer algebra program that gives all the algebraic relations between them ? I will prefer singular if it has this component.
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1answer
36 views

Quadratic graph / standard form

If I draw a graph of the quadratic $x^2-9=0$, I have a parabola with roots $x=3$ and $x=-3$ and a vertex of $(0,-9)$ with the parabola opening upwards as $a$ is positive in the standard quadratic form....
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4answers
121 views

Show that $\mathbb{F}[x^2,y^2,xy]$ is not polynomial

$\mathbb{F}[x^2,y^2,xy]$ is the polynomials in two variables whose terms all have even degrees. Of course, this generating set $x^2,y^2,xy$ is not algebraically independent, but I need to show that no ...