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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15 views

Property for nondegenerate polynomials

Let $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ be a polynomial, if the critical locus of $f$ is some isolated points or $\emptyset$, there may exsit a sequence $\{u_n\}$ to $\infty$(i.e. $\lim_{n\...
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3answers
90 views

Factorization of polynomials over $\mathbb{Z}_3$

I have been given these two polynomials $$f(t)=t^3+2t+1 \text{ & }g(t)=t^3+t^2-t+2$$ the problem says, decide if both factorization fields are isomorphic. For the second polynomial I got that $$g(...
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1answer
26 views

Berlekamp's algorithm over $Z_3$ [closed]

Using the algorithm of Berlekamp decomposed into irreducible factors polynomials: $(1)\ \ x^5 + 3x^3 + 2x + 1$ over $Z_3$ and $(2)\ \ x^4 + 2x + 1$ over $k$, $|k|=4$
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0answers
38 views

irreducibility of monic polynomials over Z [closed]

Statement : Monic polynomials irreducible over Q are irreducible over Z. Where the polynomials belong to Z[x]. How to prove or disprove the statement. It seems like the converse of gauss lemma ...
6
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1answer
90 views

Restricted equality involving prime numbers

Given three real numbers such that $a + b + c = 0$, it can be proved that \begin{align*} \frac{a^{5} + b^{5} + c^{5}}{5} & = \frac{a^{3} + b^{3} + c^{3}}{3}\cdot \frac{a^{2} + b^{2} + c^{2}}{2}\\ \...
1
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0answers
48 views

Trigonometric Roots of a Polynomial

After wondering on this question, I wondered how would you be able to find the roots of a polynomial, in the form $y=x^3+ax^2+bx+c$ if they are the sums of cosines? I'm wondering if it can, too, be ...
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2answers
65 views

Proof that $\overline{P(z)} = P(\overline{z})$ for polynomial $P$ with real coefficients

Let $$ a_0, a_1, a_2, a_3, \ldots , a_n \quad (n \ge 1)$$ denote real numbers, and let $z$ be any complex number. With the aid of $$ \overline {z_1 +z_2+ \ldots +z_n} = \overline z_1 +\overline z_2+ \...
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0answers
24 views

How to show that $\frac{((1+T)^{p^n-1}-1)}{((1+T)^{p^e-1}-1)}$ is a polynomial?

I am currently stuck at computing the polynomial (with $0<e<n$): $$\frac{((1+T)^{p^n-1}-1)}{((1+T)^{p^e-1}-1)}$$ From the context I already know that it should be a distinguished polynomial, as ...
1
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1answer
27 views

What is a polynomial with infinite number of terms?

My instructor commented that a structure function $\phi(G)$ of a graph is a polynomial if a finite number of terms. So what is the thing with infinite number of terms? Why not polynomial?
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1answer
50 views

Precision in Cubic spline interpolation

I am working on cubic spline interpolation with set of data points from CAD with following steps: Form piecewise spline equations between points. cubic equation : $ ax^3 + bx^2 +cx + d = P(x) $ ...
0
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2answers
68 views

proving no real roots exist

Prove that $x^8-x^7+x^2-x+15$ has no real roots. I did it by first assuming it has real roots and then applying Descartes rule of signs. We find that if there are any real roots, they all must be ...
0
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2answers
62 views

Finding a point on a polynomial function where there are 3 points where the line y=mx +c is touched

$f(x)=x^6 +4x^5 -6x^4-32x^3+ax^2$ $y=m\cdot x+c$ where $m$ is the gradient and $c$ is the $y$-intercept The question is that the function $f(x)$ has three $x$ values where it touches the tangent ...
0
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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
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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.
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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 ...
2
votes
2answers
55 views

Number of solutions a polynomial can have as a function of the field?

Is there any limitation (upper bound) for number of solutions of polynomial equations? Having a background in engineering, my knowledge of higher algebra is rather limited, but I do know of ...
1
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0answers
74 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 $ ,...
0
votes
0answers
41 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
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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 ...
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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.
2
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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, ...
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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: $$\...
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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 ...
7
votes
3answers
550 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 ...
3
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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 ...
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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 ...
9
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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 ...
0
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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 ...
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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
45 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 $\...
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1answer
97 views

Luröth's Theorem

I've been struggling trying to understand the Jacobson's Basic Algebra vol. II proof of the Luröth's theorem. Let $K$ be a field, $K(X)$ the field of rational fonctions and take $L$ to be a sub-...
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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))^\...
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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 ...
4
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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 $-2\cos\left(\frac {4\pi}{9}\right)$ ...
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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
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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 ...
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 ...
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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 $I$...
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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 ...
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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?
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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?
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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},~~...
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\...
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)...
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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 ...
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
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
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
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 ...