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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2
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1answer
77 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 ...
1
vote
1answer
98 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
votes
1answer
119 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 + ... + ...
1
vote
1answer
77 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. ...
3
votes
1answer
200 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?
1
vote
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 ...
1
vote
1answer
88 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 ...
-1
votes
2answers
115 views

Show that a function is a solution to differential equation

I have a homogenous differential equation $a_0 y'' + a_1 y' + a_2 y = 0$ I know that $\lambda_0$ is a double root in characteristic polynomial. Now I have to show that $y(t) = t e^{\lambda_0 t}$ is ...
5
votes
1answer
151 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 ...
1
vote
1answer
77 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 ...
0
votes
1answer
39 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 ...
2
votes
3answers
197 views

Find $p$ for which all solutions of system/equation are real

There is system of $5$ equations $$ a+b+c+d+e = p; \\ a^2+b^2+c^2+d^2+e^2 = p; \\ a^3+b^3+c^3+d^3+e^3 = p; \\ a^4+b^4+c^4+d^4+e^4 = p; \\ a^5+b^5+c^5+d^5+e^5 = p, \\ \tag{1} $$ where $p\in\mathbb{R}$. ...
0
votes
0answers
111 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 ...
1
vote
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 ...
4
votes
2answers
94 views

Solution of $Ax^5+Bx^3=C$

I have to find the positive solution of the type $Ax^5+Bx^3=C (A,B,C>0)$. It is well known that a polynomial of degree greater than $4$ does not admit an expression for the roots but I hope :D In ...
0
votes
0answers
49 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 ...
3
votes
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 $c,d \in \mathbb{Z}$.
1
vote
1answer
49 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 ...
1
vote
0answers
45 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 ...
-3
votes
2answers
184 views

Sufficient and essential condition for polynomials $P$ and $Q$ to satisfy $P(\sin x)= Q(\cos x)$

The famous identity $\sin^2 x+\cos^2x =1$ can be written as follows: The polynomials $P(x)=x^2$ and $Q(x)=1-x^2$ satisfy $$P(\sin x)= Q(\cos x),\quad \text{for all }x\in\mathbb R$$ What are ...
1
vote
1answer
66 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 + ...
0
votes
1answer
124 views

GCD of a bivariate polynomial and its partial derivative..

I am stuck in the following question :- $f(x, y)$ is a bivariate polynomial with coefficients in $Z$. We have to show that $deg(GCD(f, f_y)) > 0$ iff $deg(GCD(f, f_x)) > 0$.(Here $f_x$ denotes ...
1
vote
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)
2
votes
1answer
60 views

Roots of $z^5 (z − 2) = w $ in Unit disk

The Q is following : Prove that for each w in the unit disc $D(0, 1)$, the equation $z^5 (z − 2) = w $ has exactly five solutions in the unit disc counted with multiplicity. My Approach : let $f(z) ...
0
votes
1answer
186 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 ...
2
votes
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 ...
5
votes
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 ...
3
votes
2answers
55 views

Elementary bound theorem of a monic real polynomial

An elementary bound theorem on the roots of a real monic polynomial states that $$M := \operatorname{max} (1, |a_0| + \cdots + |a_{n-1}|) := \operatorname{max} (1, B)$$ is an upper and lower ($-M$) ...
0
votes
0answers
292 views

Formula for finding the Roots of a cubic polynomial and nature of roots depending on the discreminant

I am trying to find the roots of a cubic polynomial in variable $r$: $ar^3 - r^2 +2mr -P^2=0$, $P$, $m$ and $a$ are constants here. I know that the discriminant of this polynomial for cubic roots is: ...
7
votes
1answer
82 views

GCD of two polynomials in $ \mathbb{Z} [X]$

Let $(P,Q) \in ( \mathbb{Z} [X])^2$, such that $P$ and $Q$ don't have a common complex root, show that the sequence $\gcd(P(n),Q(n))_{n\ge0}$ is periodic. It seems to be a hard problem, please help. ...
2
votes
1answer
84 views

If $P''(x)\mid P(x)$ then $P(x)$ has all roots real or less than $3$. [closed]

Let $P(x)$ be a polynomial of degree $n$ with real coefficients, such $P(x)$ has more than $3$ real roots. Assume that $P''(x)\mid P(x)$. Prove that $P(x)$ has $n$ real roots.
1
vote
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), ...
3
votes
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: ...
0
votes
0answers
38 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 ...
3
votes
3answers
84 views

$P,Q$ are polynomials with real coefficients and for every real $x$ satisfy $P(P(P(x)))=Q(Q(Q(x)))$. Prove that $P=Q$

$P,Q$ are polynomials with real coefficients and for every real $x$ satisfy $P(P(P(x)))=Q(Q(Q(x)))$. Prove that $P=Q$. I see only that these polynomials are same degree
2
votes
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.
3
votes
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
1
vote
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 ...
0
votes
0answers
28 views

Finding polynomial generators in a subspace

$S$ is a subspace $S= \{p\in P_3|~\text{$i\in\Bbb C$ is root of $p$}\}$. So the question at hand is how do you find the system of generators for the subspace knowing that $x$ is $p$'s divisor? ...
0
votes
0answers
40 views

linear independence on polynomials

Suppose that $p_0,p_1,\dots,p_m$ are polynomials in $p_m(\Bbb F)$ such that $p_j(2)=0$ for each $j$. I want to prove that ($p_0,p_1,\dots,p_m)$ is not linearly independent in $P_m(\Bbb F)$ Now this ...
0
votes
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 ...
0
votes
0answers
25 views

Product of monic polynomials in finite fields

I am trying to show that the product of monic polynomials of degree $n$ in $\mathbb{F}_p[T]$ is given by $\prod_{i=0}^{n}(T^{p^n}-T^{p^i})$. I tried generating function but with no luck. Any hint?
-1
votes
1answer
67 views

Show any straight line is irreducible

Show that any straight line in $\mathbb{F}^{n}$ is irreducible, where F is an infinite field. I know V($ax+b$) would be a variety that represents any straight line and then V is irreducible if I(V) ...
2
votes
1answer
28 views

Determining whether two 2D polynomial curves are everywhere close to each other

Let's say we have two curves $P(t), Q(t): [0, 1] \to \mathbb{R}^2$. $P_x(t), P_y(t), Q_x(t), Q_y(t)$ are all polynomials of some degree $n$. We can further restrict this to Bernstein basis polynomials ...
0
votes
0answers
17 views

Decomposable polynomails of 2 degree in finite space $\mathbb{F}[X]$

How can I show that there is a decomposable polynomial of second degree in a finite space $\mathbb{F}[X]$? I tried contrapositive proof but I got stuck. That made me think that maybe I should go for ...
0
votes
2answers
40 views

solving the equation where two variables are used

Solve the equation $$\frac{x}{x-a} + x = \frac{b}{b-a}+ b$$ The equation doesn't make sense. Should we take the LCM .only one equation and two variabkes are given
1
vote
1answer
61 views

Can a quartic equation be reduced to a cubic/quadratic knowing that two roots are real?

I have a quartic equation that is the determinant of a 4-by-4 matrix that looks like: $det(M-\lambda I) = det \left( \matrix{m_{11}-\lambda & m_{12} & m_{13} & 0 \\ m_{21} & ...
4
votes
2answers
181 views

Polynomials and partitions

There is a question I have based on the fact: If you take a quadratic polynomial with integer coefficients, take the set $\{1,2,3,4,5,6,7,8\}$, make a partition $A=\{1,4,6,7\}$, $B=\{2,3,5,8\}$, and ...
2
votes
1answer
118 views

Finding normal basis for GF(q^m) over GF(q)

Could you kindly explain, how can one find a normal basis for GF$(3^6)$ over the GF$(3^2)$? As I understood, I should start with finding the polynomial in a form $$a(x^2) + (a^9)x + a^{81},$$ which ...
0
votes
2answers
46 views

Factor the polynomial $x^4 + 2x − 4$ in $\mathbb{Z}_5[x]$.

I'm confused as to how this is different from factoring in the reals? Would I start this by writing $x^4+2x-4 \equiv 0 \pmod 5$? What changes?