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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0answers
7 views

Transforming a polynomial sum using a series expansion (BCH codes)

In my study of BCH codes I've come across the following equation (the "key equation"): $$ \Omega(x) \equiv \Lambda(x)S(x) \mod x^{n-k} \tag{1} $$ Where the two terms on the right are defined by: $$ ...
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1answer
67 views

The roots of a polynomial are a continuous function of the coefficients

What does the following mean? The roots of a polynomial are a continuous function of the coefficients Please guide me with an example.
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1answer
18 views

Transformation in algebra?

So im doing transformations and for example if we were to write: $$y = x^3$$ and then shift it to the left by $4$ it would be written as $$y = (x - 4)^3$$ so how come in another problem I have, ...
3
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1answer
64 views

Hermite Polynomials

How can I prove that $\int^{\infty}_{-\infty} x^2 e^{-x^2} H^2_n(x)\ dx=2^nn!(pi)^{1\over2} $ My idea: $H_0(x)=1, \ H_2(x)=4x^2-2$ $4x^2=H_2(x)+2H_0(x)$ $x^2=$$1\over4$$(H_2(x)+2H_0(x))$ so ...
1
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1answer
48 views

Can the root locus of a minimum phase plant become unstable?

I have a discrete system for which the root locus equation is given as: $$A(z) + K\cdot B(z) = 0$$ They are such that $A(0) = 1, B(0) > 0$, and $K>0$. $\frac B A$ is minimum phase and a ...
2
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3answers
55 views

Trigonometric polynom

Prove that $$\cos\frac{\pi}{7},\cos\frac{3\pi}{7},\cos\frac{5\pi}{7}$$ roots of polynomial $8x^3-4x^2-4x+1=0$ I'm confused, what can i do with $\frac{\pi}{7}$
2
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2answers
45 views

Prove that a periodic nested radical converges to the largest real root of the corresponding polynomial

Let $a_k$ be a sequence of fixed positive integers, $k \in [1,n]$. Consider the periodic nested radical: $$x=\sqrt{a_1+\sqrt{a_2+\cdots+\sqrt{a_n+x}}}$$ We can transform this nested radical into the ...
0
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1answer
28 views

$Q(a,i)$ is isomorphic to a quotient of $ \mathbb Q[X,Y]$

Let $a\in \mathbb C$ be a 3rd root of 2, i.e. $a$ has minimal polynomial $X^3-2$ over $ \mathbb Q$. Claim: $ \mathbb Q[X,Y]/(X^3-2,Y^2+1) \cong \mathbb Q(a,i)$ How do I see this, do I need to ...
1
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1answer
38 views

Can the zeroes of a multivariate $p$-adic polynomial be bounded?

Multivariate real polynomials, as opposed to multivariate complex polynomials, can have bounded zero sets, i.e. $x^2+y^2-1$ in $\mathbb{R}^2$. This fails in $\mathbb{C}^n$ because $\mathbb{C}$ is ...
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2answers
53 views

legendre Polynomial

prove: $\int_{-1}^1 (x^2P_{n+1}(x)+P_{n-1}(x))\ dx$=$2n(n+1)\over {(2n-1)(2n+1)(2n+3)}$ I think to use the formula: $ (n+1)P_{n+1}(x)=(2n+1)xP_n(x)-nP_{n-1}(x)$ Then multiply the L.H.S. and ...
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1answer
47 views

legendre Polynomial

Find: $\int_{-1}^1 P_5(x)+P^2_4(x)\ dx$ my answer: $\int_{-1}^1 P_5(x) \ dx +\int_{-1}^1 P^2_4(x)\ dx$ $\int_{-1}^1 P_5(x) \ dx$ + $2\over2(4)+1$ $\int_{-1}^1 P_5(x) \ dx$ + $2\over9$ but what ...
5
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0answers
54 views

“gapped” polynomial leads to ring-shaped roots

Given a polynomial $$P(z)=\sum_{n=0}^N a_n z^n$$ with real coefficients distributed as a gaussian curve $a_n=\frac{1}{\sigma\sqrt{2\pi}}e^{(n-b)^2/2\sigma}$ ($b>0$). The sum of all the polynomial ...
0
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2answers
45 views

Prove that there is no irreducible polynomial of degree 2 in $\mathbb C[x]$.

Prove that there is no irreducible polynomial of degree 2 in $\mathbb C[x]$. I know this result is true from the Fundamental Theorem of Algebra, which states that any polynomial in $\mathbb C[x]$ ...
2
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1answer
70 views

Find the least positive integer $n$ for which $n^5+3$, $(n+1)^5+3$ cease to be coprimes

The integer polynomials $f(x)=x^5+3$, $g(x)=(x+1)^5+3$ are relatively prime (as polynomials) but this does not necessarily imply that their values $n^5+3$, $(n+1)^5+3$ (for various values of the ...
3
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1answer
26 views

$P(x)=x^3-y^2+y+1$ - Roots in $\mathbb{Q}[y]$

I would like to show that $P(x)=x^3-y^2+y+1$ doesn't have roots as a polynomial with variable $x$ and coefficients in $\mathbb{Q}[y]$ (i.e. $\mathbb{Q}[x,y]=\mathbb{Q}[y][x]$). Could anyone solve ...
1
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1answer
29 views

Ireducible polynomial over $\mathbb Z_4$

How can you prove that $f(x)=X^2+1$ is ireductible over $\mathbb Z_4$, the quotient ring? We know that $\mathbb Z_4$ admits divisors of $0$, as $2*2=0$, so any elemanary approach using $h\times g=f$ ...
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1answer
26 views

What is this ideal equal to? What is it called? “composition ideal in $R[X]$”

Let $R$ be a ring and $f(X)=f_0+f_1X+\dots +f_n X^n\in R[X]$. Define $f(J) \equiv f_0 + f_1 J + \dots + f_n J^n$ where $J^k$ is the $k$th power ideal, and $A + B = \{a + b : a \in A, b \in B\}$. ...
3
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0answers
39 views

solution of the quartic

Given the equation:$$3x^4-25x^3+50x^2-kx+12=0$$ has two roots whose product is $2$. It was given to find the value of k and all the roots of the equation.This is how I solved it: Let the roots be ...
0
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1answer
40 views

Show that the polynomial $P = x^8-6x^3+ 2x^2+2$ is irreductible in $\mathbb{Q}[x]$

Show that the polynomial $P = x^8-6x^3+ 2x^2+2$ is irreductible in $\mathbb{Q}[x]$. Is is possible to use Eisenstein's criteria? Otherwise, is anyone could help me to solve it?
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0answers
18 views

Decomposition of a polynom in a field extension

Let $K$ a field, $P \in K[X]$ irreducible of degree $n$, $L$ an extension field of $K$ with degree $m$ and $d=gcd(m,n)$. I want to show that for every $Q$ irreductible factor of $P \in L[X]$, ...
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1answer
16 views

Factoring a polynomial to get its zeros

While studying about sums and products of roots of polynomials, I found this on the web: We can take a polynomial, such as: $$f(x) = ax^4 + bx^3 +\dots$$ And then factor it like this: ...
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0answers
50 views

Comparison between roots of two polynomials

Let $m,n,p$ be natural number greater than $2$. Consider $$f(x)=(x-p+1)(x-m+1)(x-n+1)-x(2x-m-p+2)$$ We also have $g(x)$ which is obtained by changing $m$ to $m+1$ and $n$ to $n-1$ in $f$, i.e. ...
1
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0answers
33 views

How to constrain the linear least squares fit of a quadratic polynomial with known constraints

How to constrain this fit I have some function , $f(x) = a x^2+b x+c$ , with the constraints $a<0$ and $c = \frac{b^2}{4a}+\frac{1}{2}ln(\frac{-a}{\pi})$ I have measured $f(x)$ for some $x$. Can ...
1
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2answers
33 views

Prove that Every element in $F(c)$ can be written as $r(c)$ for some $r(x)$ of degree $< n$ in $F[x]$.

Let $p(x)$ be an irreducible polynomial of degree $n$ over $F$. Let c denote a root of $p(x)$ in some extension of F. Prove that Every element in $F(c)$ can be written as $r(c)$ for some $r(x)$ of ...
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0answers
27 views

Inequality for general polynomials.

This may be a bit too general for anyone to give much, but nonetheless, I'd like to see if anyone has any insight. What conditions would be necessary for the following to hold: ...
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0answers
17 views

Proof of coppersmith's theorem

Let $N$ be positive integer number and $f\in Z_N[x]$ be a monic polynomial with degree $d$. Then, there is a polynomial to find all $x\in Z$, $|x|<B$ and: $f(x)=0 (\bmod N)$ such that ...
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0answers
21 views

Uniformly Random Polynomial

Hypothesis: all the polynomials and values are defined over a finite field $\mathbb{F}_p$ where $p$ is a large prime number. My question is related to information security. Assume (in a protocol) ...
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0answers
38 views

Proof that $\mathrm{Aut}(K\left[ \sqrt[m]{g \in K} \right])/\mathrm{Aut}(K) = \mathbb{Z}_w, w | g $

Let $K$ be an extension field of $\mathbb{Q}$. That is $$ K = \mathbb{Q}[r_1,\ldots,r_k ]$$ I am considering $\mathrm{Aut}(K)$ which is the group of field automorphisms of $K$ and I wish to show ...
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0answers
76 views

Fixed point of a polynomial mapping - what's the relation between the two views

Let $\sigma : \Bbb{C}^3 \to \Bbb{C}^3$ be a polynomial mapping. Let $P:= \Bbb{C}[x,y,z]$ denote the space of polynomial in 3 variables. Then $\sigma$ induces a (linear) mapping $\tilde{\sigma} : ...
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0answers
26 views

Help with deriving Newton's Identities

I am trying to derive Newton's identities for symmetric sums, namely in the case where $k > n \geq 1$ \begin{equation} \sum_{i=k - n}^k (-1)^{i + 1}S_{k-i}P_i = 0 \end{equation} where $S_k$ is the ...
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3answers
70 views

Polynomial degree [closed]

Consider this equation: $9(x-0.4)^4+2$. I just want to confirm that this is a 4th degree polynomial.
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0answers
48 views

Is the set of all polynomials mapping $\mathbb{R}$ to $\mathbb{R}$ a closed set?

I know that by Weierstrass Approximation, P([0,1],R) is dense in C([0,1],R). But what about the case for P(R,R). Is it a closed set? I can't find any hint anywhere.
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1answer
49 views

Can anyone teach me how to answer this question??? [closed]

$\frac{x}{1-x} = \frac{{x}^{a_1}}{1-x^{d_1}}+\cdots+\frac{{x}^{a_n}}{1-x^{d_n}}$ for all $x\neq 1$.Then show that $d_1,\cdots,d_n$ cannot be all distinct.Also note that $a_1,\cdots,a_n \in \mathbb{N}$ ...
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0answers
23 views

Avg. of Roots of Polynomial

I have recently proven that the average of the roots of any polynomial will yield that polynomial's vertex. In more general terms, if $f(x) = c_0x^n+c_1x^{n-1}+..c_n$ with roots $r_1, ..., r_n$, and ...
1
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1answer
40 views

Polynomial irreducibility and perfect square.

Hi guys I am trying to show the following statement and would appreciate if people can take a look and comment if I am on the right track. $g(x,y)=y^2-p(x)$ is irreducible if and only if $p(x)$ is ...
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2answers
29 views

Coprime polynomials.

Let f(x) be a nonzero polynomial. Show that there exists a polynomial g(x) with f(x) g(x) ≡ 1 (mod p(x)) if and only if gcd (f(x), p(x)) = 1. What is this math question asking us to prove? Is ...
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1answer
35 views

Self adjoint differentiation operator on vector space of polynomials and inner products.

Let $n \in \mathbb{N}$. For the following, consider the vector space of polynomials of degree less than or equal to $n$ with complex coefficients, $\mathcal{P}_n(\mathbb{C})$. a) If we equip the ...
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4answers
387 views

Factorization of polynomials with degree higher than 2

I need help to factorize $x^4-x^2+16$. I have tried to take $x^4$ as $(x^2)^2$ and factorize it in the typical way of factorizing a quadratic expression but that did not help. Can someone help me to ...
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1answer
33 views

Simple roots of a polynomial are smooth functions with respect to the coefficients [duplicate]

How can we prove by implicit function theorem that: Question: The simple roots of a polynomial are smooth functions with respect to the coefficients of the polynomial?
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0answers
19 views

How to show map t nilpotent?

What is the best what to show that $t$ is nilpotent for the following question? "Let $V$ be the set of all rational polynomials of degree at most two and let the linear map $t : V \to V $ be given by ...
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1answer
47 views

Contradiction in polynomial equation.

I am getting a contradiction in solving this question. Solve the equation $$(x^2-3x+3)^2 -3(x^2-3x+3) +3 = x$$ Here is what I did. $$f(x) = x^2-3x+3$$ Then, the equation becomes, $$[f(x)]^2 ...
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1answer
43 views

Simple question about finding roots of a polynomial

What am I doing wrong here? This is the denominator of one of my problems and I need to find the roots, so: $6i-z^2+1 \to z=\sqrt{1+6i}$ and $z=-\sqrt{1+6i}$ $\therefore$ ...
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0answers
15 views

How to estimate the size of each coefficient of the result of a polynomial multiplication?

Let $a$ and $b$ be two polynomials of degree $D$. Moreover, suppose the $i$-th coefficient of $a$ and $b$ lies in the range $\left[0, 2^{n_i}\right)$. Now, let $c = a \cdot b$. This implies that ...
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5answers
64 views

How to verify that one of equations in a polynomial system is redundant?

I know that system of polynomial equations $$ p_1(x_1,\dots,x_n)=0,..., p_N(x_1,\dots,x_n)=0 $$ has infinitely many solutions. I computed some of them numerically and notices that they always satisfy ...
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1answer
25 views

Seeking explanation for an identity in: “Polynomials such that roots=coefficients”

In the following thread here: http://math.stackexchange.com/users/66096/legranddodom, LeGrandDODOM stated the following identity (2): $ \sum\limits_{1 \le i < j \le n} z_i z_j = z_2 $ for: ...
2
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1answer
26 views

Find all the roots of this polynomial

I'm currently stuck with the following problem: Find all the roots of the equation $$1-\frac{x}{1}+ \frac{x(x-1)}{2!}-...+(-1)^n \frac{x(x-1)...(x-n+1)}{n!}=0$$ I can sort of see that the roots ...
1
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3answers
54 views

How to solve a polynomial $P(x) = 0$

At the moment we just learnt the factor theorem of polynomials and how if $x-a$ is a factor of $P(x)$, then $P(a) = 0$. We're then taught to find the roots of a polynomial its best to check the ...
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0answers
31 views

Prove Descartes' Rule of Signs for a Polynomial of a Specific Degree

I am trying to prove Descarte's rule of signs for polynomials of a specific degree. essentially i'm trying to prove that the rule of signs is true for polynomials of the form: $ ax^4 + bx^3 + cx^2 + ...
1
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0answers
19 views

Sign of bivariate polynomial evaluated over two algebraic numbers

I would like to compute the sign of a bivariate polynomial $f$ evaluated over two algebraic numbers $a$, $b$. The numbers are in "isolating interval representation" meaning that each one is defined by ...
1
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0answers
32 views

Isolating roots of polynomial system

I would like to isolate the regions which contain the roots of a system of two bivariate cubic polynomials. I thought I would project the solutions onto $x$ and $y$ axis by means of resultant ...