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Questions tagged [polynomials]

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 and checking for irreducibility.

0
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3answers
23 views

Polynomial with properties at its zeroes - Linear Algebra

Take $P_{n+1}(\Re)$ to be the space of polynomials with coefficients in the real numbers, with the degree of all polynomials $\le n+1$. I want to show that, given any $n$ points $a_1, a_2, ..., a_n \...
1
vote
2answers
24 views

How to go from polynomial of third degree to multiplication of two smaller polynomials

I have a basic calculus question which I should be able to do easily but I just can't remember how to tackle it. I'm working on my linear algebra exam and trying to find the eigenvalues for a certain ...
3
votes
1answer
42 views

Number of roots of a quadratic polynomial with coefficients in ring $\mathbb{Z_{18}}$

I am trying to solve the following problem: How many roots can a polynomial $P(x) = ax^2 + bx + c $, where $a$, $b$, $c \in \mathbb{Z_{18}}$, have? ($\mathbb{Z_{18}} = \{0, 1, 2, \ldots, 17\}$)...
0
votes
0answers
27 views

Function consisting of two square functions

In last time, I define a problem which I'm trying to solve but I don't know how can I do it. Let we say, that we have function $$ Y(u,k) = \frac{uk}{\sqrt{ (u^2k^2+uk(nx+my)+nmxy)}}. $$ As you can ...
3
votes
1answer
25 views

Showing algebraic dependence of meromorphic functions on a compact Riemann surface

I have been given the following question to do: Let $f,g$ be meromorphic functions on a compact Riemann Surface $R$. Show that there is some polynomial such that $P(f,g) = 0$ (i.e. show that any two ...
2
votes
2answers
45 views

Prove that $R$ is not a UFD [on hold]

Let, $$R=\{a_0+a_1x+\dots+a_nx^n \in \mathbb{Q}[x] : a_0 \in \mathbb{Z} , n \in \mathbb{N_0} \}$$ Prove that $R$ is not a UFD. I tried to find some irreducible element which is not prime but ...
2
votes
2answers
45 views

Find $p$ if the remainder when $(x^2 + px + 13)$ divided by $(x-2)$ is twice the remainder when it is divided by $(x+2)$.

As far as I understand, this should be a simultaneous equation, although I'm not really sure where to go beyond that. I made the remainder of $f(-2)=r$ and $f(2)=2r$, and substituted the respective ...
2
votes
1answer
34 views

How to show that $\phi: \mathbb{A}^1(\mathbb{C}) \to V(Y^2 - X^3)$ is no isomorphism.

Consider the polynomial function $$\phi: \mathbb{A}^1(\mathbb{C})\to V(Y^2 - X^3): t \mapsto (t^2, t^3)$$ Show that $\phi$ is a bijective polynomial function, but no isomorphism (= $\phi^{-1}$ is ...
5
votes
2answers
934 views

Mysterious polynomial sequence

Can someone identify this polynomial sequence? Is it known in mathematics? I'm interested in various properties of this sequence. I'd like to find $P(n)$, $n\in \mathbb{Z}^+$ \begin{align} P(0)&= ...
1
vote
1answer
39 views

Prove: $|\{(x,y)\in \Bbb F_q^2 : y^2 = Q(x) \}|=q-1$ in a finite field

Let $\Bbb F_q$ be a finite field with q elements, with q odd. For a quadratic polynomial $Q(T)=T^2+a \in \Bbb F_q[T]$, show that if $a \neq 0$, then: $$|\{(x,y)\in \Bbb F_q^2 : y^2 = Q(x) \}|=q-1$$ ...
2
votes
2answers
48 views

Question about the formal definition of a polynomial in relation to $\sin(x)$ not being a polynomial

This question has been asked before, but none of the answers seems to satisfy what I'm asking here .. So, in class, we introduced set of polynomials as a set of sequences of real numbers, where only ...
3
votes
1answer
33 views

A computational criterion of irreducibility in $\mathbb Z[X]$?

If $f\in\mathbb Z[X]$ and there are $x_1,\dots, x_n\in\mathbb Z_+$, where $n>\deg f$, such that $f(x_i)\in\mathbb P$, $i=1,\dots n$, then $f$ is irreducible over $\mathbb Z$. Because, if $\,f=g\...
4
votes
2answers
88 views

Compute $\sum\frac1{2-A_k}$ for $(A_k)$ the $n$th roots of unity [duplicate]

If $1,A_1,A_2,A_3....A_{n-1}$ are the $n^{th}$ roots of unity then prove that $$\dfrac{1}{2-A_1} + \dfrac{1}{2-A_2}+\cdots+ \dfrac{1}{2-A_{n-1}} = \dfrac{2^{n-1}(n-2) + 1}{2^n-1}$$ What I did: I ...
0
votes
1answer
28 views

Can we find $\lambda \in \mathbb{C}$ such that $\deg(\gcd(f(t-\lambda),g(t-\lambda)))=1$?

Let $f=f(t),g=g(t) \in \mathbb{C}[t]$ with $\deg(f) \geq 2$ and $\deg(g) \geq 2$. Can we find $\lambda \in \mathbb{C}$ such that $\gcd(f(t-\lambda),g(t-\lambda))$ is of degree $1$? Please notice ...
1
vote
0answers
64 views

If $\langle A,B \rangle =\langle C,D \rangle$, then $\langle A-\lambda,B-\mu \rangle = $?

Let $A,B,C,D \in \mathbb{C}[x,y]$, with $\deg(A),\deg(B),\deg(C),\deg(D) \geq 1$. Assume that the ideal generated by $A$ and $B$ equals the ideal generated by $C$ and $D$, namely, $\langle A,B \...
0
votes
2answers
26 views

Why is a polynomial with infinite zeropoints the zeropolymomial? [duplicate]

This was given us as a fact, but why is this true? The zeropolynomial is the polynomial where all the coefficients are equal to $0$ if $R(x)$ is a polynomial over $\mathbb{C}$ and every $x\in \mathbb{...
0
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0answers
12 views

Polynomial with 2 variables and complex coefficients, if $P(k,l)=0,\forall k,l\in \mathbb{N}_0$ then $P=0$

This is an excercise with 4 parts, I have difficulties with the third part. I will show you the results of the first two parts. I Need help with an inductionproof. $(i)$Let $P(x,y)$ be a polynomial ...
1
vote
1answer
34 views

Galois field inverse

I'm trying to calculate the inverse of $x^{7}+x^{5}+x^{4}+x^{2}+x+1$ over $\mathbb{Z}_{2}[x]/(x^{8}+1)$. I suspect there is something fundemental I'm misunderstanding about this process. I'm doing ...
3
votes
5answers
43 views

Find all the solutions of $z^2-(1+3i)z-8-i=0$

I am stuck on a problem and I was hoping someone could tell me what I am doing wrong. I want to find all the roots of: $$z^2-(1+3i)z-8-i=0$$ There are two ways I tried to approach this. ...
1
vote
0answers
77 views

Show that $f(x)-P(x)=\frac{x^4-1}{4!}f^{4}(c) $

If $P(x)$ is a unique cubic polynomial for which $P(x_0)=f(x_0),P(x_2)=f(x_2),P^{'}(x_1)=f^{'}(x_1),P^{''}(x_1)=f^{''}(x_1)$,$f(x)$ is a given function differentiable $4$ times. Show that $f(x)-P(...
0
votes
1answer
21 views

k non-attacking rooks on a chessboard with forbidden positions

[Barbeau, Polynomials, page 8] I am trying to understand the equation shaded in the extract below: Unfortunately the wikipedia entry only has complete boards (all squares allowed) Now for some ...
3
votes
1answer
39 views

Why $\mathbb{C}(f(t),g(t))=\mathbb{C}(t)$ implies that $\gcd(f(t)-a,g(t)-b)=t-c$, for some $a,b,c \in \mathbb{C}$?

Assume that $f(t),g(t) \in \mathbb{C}[t]$ satisfy the following two conditions: (1) $\deg(f) \geq 2$ and $\deg(g) \geq 2$. (2) $\mathbb{C}(f(t),g(t))=\mathbb{C}(t)$. In this question it was ...
4
votes
2answers
60 views

$f^n+g^n=h^2\implies f,g,h$ all constant

Let $k$ be a field with char$(k)=0$ and suppose for $f,g,h\in k[x]$ having gcd$(f,g,h)=1$ and $n\in\mathbb{N}_{\geqslant4}$ it holds that $f^n+g^n=h^2$. I want to show that $f,g,h$ are all constant. ...
1
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0answers
43 views

Wanted to check if this proof is correct [Polynomials, 1st year university]

WTP: If $P\in {\Bbb C[x]}$ has a leading coefficient $a_n$, then P factorises completely into linear factors in $\Bbb C$, $$P(x) = a_n(x - \alpha)(x - \alpha_1)(x - \alpha_2)\ldots(x - \alpha_n)$$ ...
0
votes
1answer
19 views

If $f(x)$ is irreducible in $ \mathbb z [ x]$ , then for all primes $p$ the reduction $f'(x)$ of $f(x)$ modulo $p$ is irreducible in $F_p[x]$.

If $f(x)$ is irreducible in $ \mathbb z [ x]$ , then for all primes $p$ the reduction $f'(x)$ of $f(x)$ modulo $p$ is irreducible in $F_p[x]$. Is the statement ...
0
votes
1answer
39 views

Solve cubic polynomial [duplicate]

How can I solve this third-degree polynomial? I want to solve it for $y$. $x=y^3+y-9$ I can simplify it to $x+9=y(y^2+1)$ but I don't get any further.
1
vote
1answer
49 views

Quotient $\mathbf{F}_3[X]/(X^5+1)$

Factor $X^5+1\in\mathbf{F}_3[X]$ into irreducibles. What does the quotient $\mathbf{F}_3[X]/(X^5+1)$ look like? Since $-1$ is a zero, we divide $X^5+1$ by $X+1$ using long division, to obtain $X^5+1=(...
0
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0answers
29 views

Can I solve a system of high-degree polynomial equations with the Gröbner basis method?

I have two equations, two unknown (ISP, ch) and one parameter ($n_C$). My aim is to find the values of the unknown according to the parameter. The original rational equations are here : https://...
0
votes
0answers
39 views

Is it true that he polynomial $\frac{x^p- 1}{x-1}$ ($p$ is prime) is irreducible in $\mathbb{F}_2[x]$ iff $p$ is prime?

Is it true that the polynomial $\frac{x^p- 1}{x-1}$ ($p$ is prime) is irreducible in $\mathbb{F}_2[x]$ iff $p$ is prime? I know it will be true in $\mathbb Q[x]$. Can anyone please help me to ...
1
vote
1answer
46 views

$x^{2n} + x^{2n-1} + x^ {2n-2} +\ldots+ x + 1$ is irreducible for any $n\in \mathbb N$ in $F_2[x]$. True or false?

Will the polynomials of the following set $A$ be irreducible in $F_2[x]$? $A = [x^{2n} + x^{2n-1} + x^ {2n-2} + \ldots+ x + 1 : n\in \mathbb N]$ Can anyone please give me hints how to proceed? ...
-2
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0answers
28 views

Product of the roots- polynomials [on hold]

Why is the product of the roots multiplied with a minus if n is odd?
0
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0answers
40 views

Sum & Product of Two Algebraic Numbers [duplicate]

If you have algebraic numbers $x$ and $y$, and you know the polynomials of least degree of which each is a solution (written as a vectors of coefficients), then how is the vector of coefficients of ...
1
vote
0answers
18 views

Polynomial approximation of a function in a chosen interval

I don't know if this should be in the math forum or the computer science one, but I think it is more relevant in the math section. I would like to approximate nonlinear functions typically used in ...
1
vote
0answers
56 views

Differentiate a simple root of a polynomial curve

Suppose $f(t, z) = z^n + a_{n-1}(t) z^{n-1} + \dots + a_0(t)$ is a polynomial curve in the sense: each coefficient $a_j: I \to \mathbb C$ is a $C^{\infty}$ function over some compact real interval. ...
1
vote
1answer
46 views

Can we find all the irreducible polynomials of $F_2[x]$?

Can we find all the irreducible polynomials of $F_2[x]$ of a degree $n$? Is the number of irreducible polynomial of $F_2[x]$ Infinite? I was to find if there is any degree $n$ such that there is no ...
0
votes
2answers
21 views

Polynomial Function given roots and a point

A 3rd degree polynomial has roots at x=-2i and x=5. The y-intercept is (0,25). Write an equation for this function in factored form with real coefficients. My first hack is to say that in factored ...
1
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0answers
38 views

Concerning $\frac{\mathbb{C}[x,y,z]}{\langle x^2-1, yz \rangle}$

Let $R=\frac{\mathbb{C}[x,y,z]}{\langle x^2-1, yz \rangle}$. For convenience, I will write $x,y,z \in R$ instead of $\bar{x},\bar{y},\bar{z}$. In $R[X,Y]$, take $A=(x+iy)X+yY$, $B=yX+(x-iy)Y$, $C=zX$...
0
votes
1answer
24 views

Roots of sparse “quadratic-like” polynomial.

So I know about this question and I've seen papers like this and this. But the former isn't exactly what I want and the latter two papers are too deep and I'm lazy and I wanna quick-and-easy answer ...
5
votes
1answer
79 views

What is an irreducible element in $\Bbb{Z}_6[x]$?

What is an irreducible element in $\Bbb{Z}_6[x]$? This was a problem on our final and no one knew how to solve it. Does anyone have a method for solving this?
1
vote
1answer
26 views

Let $v \in \operatorname{Im}(p)$. Compute $p(v)$.

Let $B = (1, X, X^2)$ be an ordered basis for $\Bbb R_2[X]$ and $p ∈ \mathcal{L}\big(\Bbb R_2[X]\big)$ be the linear map defined by $p(1) = \frac{1}{3}(2 − X − X^2)$, $p(X) = \frac{1}{3}(−1 + 2X − X^2)...
1
vote
0answers
55 views

Approximate expectation of a polynomial of independent Bernoulli variables

$X_{1}, ... , X_{K}$ are $K$ independent Bernoulli random variables $c_{jk}$ are constants between 0 and 1. $p_{k} = P(X_{k} = 1)$ are known. Is it possible to simplify the following expectation? $$\...
6
votes
1answer
51 views

prove a polynomial has at least 2n-1 distinct real roots

If $P(x)$ is a real polynomial has $n$ distinct real roots in $(1,+\infty)$. Set: $$Q(x)=(x^2+1)P(x)P'(x)+x(P^2(x)+P'^2(x))$$ How to prove $Q(x)=0$ has at least $2n-1$ distinct real roots? I think ...
1
vote
0answers
28 views

How vanishing set irreducibility affects homogeneous polynomials

I am trying to gain some insight into how the irreducibilty/reducibility of the vanishing set of a homogeneous polynomial affects the composition of the polynomial itself, so I am looking for two ...
0
votes
3answers
36 views

Finding minimum value of $xy$ given that $1/x + 1/y =2$? [closed]

If for $x>0$ and $y>0$ we have $1/x + 1/y =2$. What can be the minimum value of $xy$? How do we do this simple question?
3
votes
0answers
61 views

Finding the cube roots of two complex numbers knowing the product of these cube roots

Let $c_E$, $c_F$, $c_G$ and $c_H$ be known real numbers. Let $x_1$, $w_1$, $x_2$ and $w_2$ be unknown real numbers. When solving a cubic equation, one could be led to solve that kind of system: \...
1
vote
0answers
17 views

Degree of resultant of polynomials in $\mathbb{Z}[Y][X]$

I am trying to write a program that computes the resultant of two polynomials with coefficients in $\mathbb{Z}[Y]$. If I know the degree of the polynomial as a polynomial in $Y$ is $k$ say, I can ...
1
vote
0answers
39 views

Show $ \sum_f \lambda(f)t^{\deg f} = \prod_f \big(1 - \lambda(f)t^{\deg f}\big)^{-1} $

So I want to show that $$ \sum_f \lambda(f)t^{\deg f} = \prod_f \big(1 - \lambda(f)t^{\deg f}\big)^{-1} $$ where the sum is over monic polynomials and the product is over all monic irreducible ...
1
vote
1answer
56 views

I want to know how to prove ideal

I'm just learning about ideal and struggling with these problem. So I want to know how to prove these. About ℚ[x] ⊂ C[X], a ∈ C, $I_a = \{f(X) ∈ ℚ[x] | f(a) = 0\}$ (1) Show $I_a$ is an ideal of $ℚ[...
0
votes
0answers
33 views

Roots of polynomials are Gaussian integers

I have got a question. I want to show the following: Let P be a normalized polynomial with integer coefficients and let w be a root of this polynomial (in $\mathbb{Q}[i]$), then w is a Gaussian ...
0
votes
0answers
31 views

Gröbner basis and generating sets [on hold]

Define an ideal $L:=\langle xz − y^2 + z, x^3 − yz^2, yz −y^2 \rangle ⊆ Q[x,y,z]$. Compute a generating set for $L \cap Q[y]$. Compute a generating set for $L \cap \langle y\rangle$. Compute a ...