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.

learn more… | top users | synonyms

4
votes
1answer
171 views

Complex solutions within unit circle

How many solutions within a unit circle $|z| < 1$ does the equation $(1 + z)^{n + m} = z^n$ have for $z$ complex, and $n$, $m$ positive integers?
12
votes
7answers
12k views

How to prove that a polynomial of degree $n$ has at most $n$ roots?

How can I prove, that a polynomial function $$f(x) = \sum_{0\le k \le n}a_k x^k\qquad n\in\mathbb N,\ a_k\in\mathbb C$$ is zero for at most $n$ different values of $x$? (Except $n=0$ where $f(x)$ is $...
3
votes
1answer
45 views

$\sin(nx)$ espansion into $n$-th grade $\sin(x)$ polynomial

Maybe this is a well-know question, anyway I haven't found an exact duplicate. It is possible to express $\cos (nx)$ as a polynomial of degree $n$ in $\cos(x)$. As stated in this answer, it is ...
2
votes
2answers
53 views

To show that $\langle x-a , y-b\rangle$ is a maximal ideal of $F[x,y]$ by showing that $F[x,y]/\langle x-a , y-b\rangle$ is a field [duplicate]

Is there any way to show that for $a,b \in F$ , the ideal $\langle x-a , y-b\rangle$ is maximal in $ F[x,y]$ , by showing that the quotient $F[x,y]/\langle x-a , y-b\rangle$ is a field ? Is the ...
0
votes
0answers
30 views

Possible root of a polynomial $p(x)=x^n+a_{n-1}x^{n-1}+…a_1x-1$ [duplicate]

Let $p$ be a real polynomial of the real variable $x$ of the form $p(x)=x^n+a_{n-1}x^{n-1}+....a_1x-1$. If $p$ has no roots in the open unit disc and $p(-1)=0$, then can we predict the other possible ...
1
vote
0answers
61 views

If $f(z) = sinz$ has infinitely many solutions , then f is constant. [closed]

Let $f(z)$ be complex polynomial. Prove that If $f(z) =\sin z $ has infintely many solutions , then $f$ is constant. I think it may be proved by Rouche theorem.
0
votes
2answers
19 views

Determine the points where polynomial function intersects logarithmic function

Given the function $n^k$ where $k$ is a constant such that $0<k \leq k_{max}$ where $k_{max}$ is the point at which $n^k$ first intersects with $log_2n$ determine: $k_{max}$ For a given $k$, the ...
1
vote
0answers
76 views

Trigonometric solution to solvable equations

The algebraic equations in one variable, in the general case, cannot be solved by radicals. While the basic operations and root extraction applied to the coefficients of the equations of degree $ 2 $ ,...
1
vote
0answers
43 views

Action of a Linear Functional on a Polynomial

I was hoping to find a good canonical reference for the mathematics behind something called the action of a linear functional $L$ on a polynomial $p(x)$ which is denoted $\langle L|p(x)\rangle$ ...
0
votes
1answer
28 views

Can we estimate $P(x)$ using $P(1)$?

Given a polynomial $P(x)$, is it possible to estimate/lower bound/upper bound the value of $P(k)$ for some $k \in \mathbb{N}$ if we know $P(1)$? We can also assume $P(x)$ has only natural ...
-1
votes
0answers
27 views

If f(x) is a polynomial whose coefficients are all +1 or -1 and whose roots are real, then degree of f(x) can be : (a)1 (b)2 (c)3 (d)4 (e)5 [duplicate]

I was able to work on 1,2 and 3 degree polynomial. But could not come to any conclusion in case of 4 and 5 degree polynomial.
0
votes
1answer
34 views

Solutions of a system of polynomial equations

I am trying to find the critical points of some functions such as $$f(x, y) = x^4 − x^2y^2 + y^3 − 18x^2 + 3y^2$$ I calculate the gradient, and then find a system of polynomial equations: $$\...
7
votes
3answers
552 views

Eigenvector of polynomial

Suppose that $T: V \rightarrow V$ is an endomorphism of the linear space V (about $\mathbb{K}$) and that $p(X)$ is a polynomial with coefficients in $\mathbb{K}$. Show that if $x$ is an eigenvector of ...
2
votes
0answers
30 views

Books about multivariate polynomials

I'm looking for a book on multivariate polynomials, preferably a monograph (could also be a chapter inside another book). I'm interested in what can be said about roots, factoring, irreducibility, ...
0
votes
0answers
13 views

fifth-degree and Bring-jerrard

According to this post , If values for the coefficients For example, consider The steps above are some of the parameters have complex value, Like the amount u ; v ; p ;.... Faced with complex values ...
10
votes
3answers
246 views

Is it true that if $f(x)$ has a linear factor over $\mathbb{F}_p$ for every prime $p$, then $f(x)$ is reducible over $\mathbb{Q}$?

We know that $f(x)=x^4+1$ is a polynomial irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$. My question is: Is it true that if $f(x)$ has a linear factor over $\...
9
votes
0answers
87 views

Integer polynomials with roots in every $\mathbb{Z}_p$ but no rational roots.

I want to find polynomials in $\mathbb{Z}[x]$ with degree as small as possible such that these polynomials have no rational roots but have a root in the $p$-adic integers $\mathbb{Z}_p$ for every ...
4
votes
2answers
88 views

How would you find the roots of $x^3-3x-1 = 0$

I'm not too sure how to tackle this problem. Supposedly, the roots of the equation are $2\cos\left(\frac {\pi}{9}\right),-2\cos\left(\frac {2\pi}{9}\right)$ and $-2\cos\left(\frac {4\pi}{9}\right)$ ...
0
votes
0answers
10 views

Prove that (x - c_j) divides q - q(c_j).

Let $c_1, c_2, ..., c_n$ are distinct scalars in field F. Choose one $c_j$ among the c's. Let q = $\prod_{i \ne j} (x - c_i)$. I am trying to prove that $(x - c_j)$ | q - q($c_j$). I have done some ...
0
votes
1answer
46 views

How to show the dimension of the vector space K[X]/fK[X]?

Let K be a field and f$\neq$0 $\in$ K[X] a polynom. a) Show that the Ring K[X]/fK[X] is a K-vector space with the dimension n=deg(f) b) f is called irreducible, if for g,h \in K[X] we have f=g*h $\...
1
vote
1answer
15 views

Multiplication of polynomials in their point value form

I've never really understood how point-value multiplication of polynomials work, so I was wondering if somebody could talk me through it with an example. Say if I was given the following two ...
-2
votes
1answer
48 views

Give two polynomials in $\mathbb Q[x]$ (of degree 2 and 3) such that their product is an irreducible polynomial in $\mathbb Q[x]$ of degree 5 [closed]

I know that $$x^k - p, \ \ \forall k>0\in N$$ is irreducible in $\mathbb Q[x]$ (Eisenstein theorem). I need two polynomials in $\mathbb Q[x]$ (one of degree 2 and another of degree 3) such that ...
0
votes
1answer
39 views

Is it possible to have a such polynomials?

An exercise asks me to write an example of such polynomials, if they exist: an irreducible polynomial of degree 5 in $\mathbb{R}[x]$. a polynomial of degree 5 in $\mathbb{R}[x]$ that has no roots a ...
5
votes
0answers
64 views

Show that $\sum_{d\mid f} \varphi(f/d) a^{|d|} \equiv 0 \pmod f$

This equation is correct when $f$ and $a$ are any integers. I want to show that this holds for $f,a\in K[x]$ where $K$ is any finite field. In the equation $\varphi(f)$ is defined as $|(K[x]/(f))^\...
0
votes
0answers
11 views

Polynomial has not roots on disc D($0$,$1$) [duplicate]

Let p($z$)=$a_n$$z^n$+..+$a_0$ with 0< $a_n$ $\le$ $a_{n-1}$ $\le$...$\le $$a_0$ .Show that p($z$) has not roots on D={ ℂ $\exists$ $z$ : $|z|<1$ }. thanks
1
vote
1answer
37 views

interpolation polynomial error

We have points $x_0=a \lt x_1 \lt x_2 ....x_n=b $ and $\;w_{n+1}(x)=\prod_{k=0}^{n}{(x-x_k)}$. Let $h=max_{j=0...n}|x_j-x_{j-1}|$ Let $f \in C^{n+1}[a;b]$ and $p_n\in \mathbb P_n$ be the ...
2
votes
2answers
136 views

Prove that $(x^2+1)\mathbb Z[x]$ is a prime ideal of $\mathbb Z[x]$, but not maximal

Prove that $(x^2+1)\mathbb Z[x]$ is a prime ideal of $\mathbb Z[x]$, but not maximal. I'm supposed to show this for my homework. My first thought is to show that $\mathbb Z[x]/(x^2+1)\mathbb Z[x]$ ...
0
votes
0answers
19 views

coefficients in Bring-jerrard form

According to this post , If values for the coefficients For example, consider The steps above are some of the parameters have complex value, Like the amount u ; v ; p ;.... Faced with complex values ...
1
vote
2answers
49 views

Not fully understanding polynomial quotient rings.

This is my (informal) understanding of a quotient ring. I understand that this is very flimsy, but I hope you can get the main idea. You have some ring $R$ and you want to quotient out an ideal $I$...
4
votes
0answers
79 views

The $2 \times 3$ matrices with rank $\leq 1$ cannot be defined by two polynomial equations

Let $X$ be the space of all ${2 \times 3}$ matrices over $\mathbb{C}$ that have rank at most 1. This is naturally a subspace of $\mathbb{C}^6.$ We can express $X$ using 3 polynomial equations, namely ...
0
votes
1answer
23 views

Polynomials bounded on integers

Let $p:\mathbb{R}\rightarrow \mathbb{R}$ be a real valued polynomial, such that for all integers $0\leq i\leq n$ we have $b_{1}\leq p(i)\leq b_{2}$. Let $k=\max_{0\leq x\leq n}|p'(x)|.$ Then for all ...
5
votes
2answers
52 views

Pell equation in ${\mathbb Q}(x)$

Is it known whether the equation $A^2-(x^2+3)B^2=1$ has a solution $A,B\in{\mathbb Q}(x)$ with $B\neq 0$ ? My thoughts : I think that there is no solution, as the fundamental solution of $A^2-(x^2+3)...
0
votes
0answers
32 views

Question about radical ideal

Suppose $\mathbb{R}[X]$ is the normal multivariate polynomial ring where $X = x_1,...x_n$. $\mathbb{R}[X]_t$ is the truncated set such that $\mathbb{R}[X]_t =\left\{f: f \in \mathbb{R}[X], \deg(f) \...
1
vote
0answers
35 views

Finiteness of solutions to system of polynomial equations $P(x)P(y)=1$ & $Q(x)Q(y)=1$

Can that finiteness be proved for polynomials $P^n\neq\pm Q^m,\quad n,m>0\;$ by known methods?
0
votes
1answer
63 views

What does the notation $O(x^n)$ mean?

I am reading a book about Padé Approximations, and I am trying to understand the following line: We denote the $[L/M]$ Padé approximant to $A(x)$ by $A(x) - P_L(x)/Q_M(x) = O(x^{L+M+1})$ where $P_L(...
3
votes
0answers
39 views

How to solve a quintic polynomial equation?

I know that not all quintics are solvable. But how do I identify the class of solvable ones?
2
votes
1answer
68 views

How can show the following function has real roots?

I was working on a problem and it reduced to solve the equation $f(a)=0$ where $$f(a)=\frac{\sum_{i=1}^{s}\sum_{j=1}^{r}(n-j+1)ja^jx_{ij}}{\sum_{i=1}^{s}\sum_{j=1}^{r}(n-j+1)a^jx_{ij}}-\frac{r+1}{2},~~...
13
votes
3answers
642 views

How would you find the exact roots of this equation?

My friend asked me what the roots of $y=x^3+x^2-2x-1$ was. I didn't really know and when I graphed it, it had no integer solutions. So I asked him what the answer was, and he said that the $3$ roots ...
0
votes
0answers
9 views

Is this implicit mapping convex?

I am interested in the convexity properties of the following mapping on the $n\times 1$ vector $x$: $$ x_{j}=y_{j}^{\beta}\left(\sum_{i=1}^{n}B_{ij}x_{i}\right)^{\alpha} $$ where $\beta>0$, $y_j\...
1
vote
1answer
538 views

Newton backward interpolation in Mathematica

I have the following task: Create a function (in Wolfram Mathematica), called $\mathrm{NewtonBackward}$[n_,x0_,h_,f_] which interpolates backwards the function $f(x)$ with nodes {x_i = x_0 + $i$$\...
1
vote
3answers
170 views

On every other finite field at least one of −1, 2 and −2 is a square, because the product of two non squares is a square

[Except on field extensions of $\mathbb{F}_2$] On every other finite field at least one of $−1$, $2$ and $−2$ is a square, because the product of two non squares is a square. I don't see why this is ...
2
votes
2answers
64 views

The expansion of $(a+b+c+d)^{20}$ [closed]

Let us consider the expansion of $$(a+b+c+d)^{20}.$$ Find: The coefficients of $a^{11}b^6c^2d$ and $a^{11}b^9$, The total number of terms of this expansion, The sum of all the coefficients. Thank ...
2
votes
1answer
34 views

General method: show subset of $\mathbb{C}$ is connected

Consider the two sets $$ A = \{z \in \mathbb{C} : |z^2 - 3| < 1\}, ~~~~ B = \{z \in \mathbb{C} : |z^2 - 1| < 3 \} $$ $B$ is connected, while $A$ is not. However, I have no idea how to prove this....
2
votes
0answers
36 views

How to define hypergeometric function ${}_1 F_1(-n+1;-n+1;z)$ for $n$ positive integer

Consider a truncated Taylor series of the exponential function to approximate $e$: $$ E(n) = \sum_{k=0}^{n-1} \frac{1}{n!} $$ I thought of computing this using the hypergeometric finite series $_1 F ...
1
vote
2answers
63 views

Finding an interval in which all the real roots of a polynomial lie

I'm making a program which uses simple bisection method to find the roots of a polynomial. For me to implement this method, I need a rough interval where it can be said with absolute certainty that ...
1
vote
0answers
38 views

fifth degree equation

The general form of the fifth degree equation to achieve Bring-jerrard form some coefficients are complex . Should they be considered only real part of the coefficients that the roots of the fifth ...
1
vote
1answer
56 views

Polynomial that is irreducible over $ \mathbb{Q} $ but reducible over every finite field [duplicate]

I want to prove that $ X^4 - 10X^2 + 1 $ is reducible in $ \mathbb{F}_p[X] $ for every prime number $ p $, but it is irreducible over $ \mathbb{Q} $. I am not sure how to approach this problem; any ...
0
votes
3answers
686 views

Methods to show linear independence of functions, polynomials

There are apparently many methods to show that a set of functions are linearly independent. Forexample, there were some cases, where I saw use of derivative. At least to prove Lgrange polynomials are ...
14
votes
6answers
2k views

Can the product of two polynomials result in a single term?

Assume that the polynomials that we multiply consist of more than one term. I don't think we can get a result containing only a single term, but I don't know how to prove it.
0
votes
2answers
80 views

Solve $z^5=-32$ and draw its solutions in complex space, then describe their characteristic geometrical property.

I'm solving past exam questions in preparation for an Applied Mathematics course. I came to the following exercise, which poses some difficulty. If it's any indication of difficulty, the exercise is ...