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Questions tagged [symmetric-polynomials]

Questions on symmetric polynomials, polynomials in several variables that are invariant under permutation of the variables.

6
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1answer
121 views

Solve the system of $3$ quadratic equations [on hold]

Consider the system of equation $${{x}^{2}}+{{(1-y)}^{2}}=a\\ {{y}^{2}}+{{(1-z)}^{2}}=b\\ {{z}^{2}}+{{(1-x)}^{2}}=c$$ Compute $x(1-x)$ in terms of $a,b,c$.
5
votes
0answers
44 views

Symmetric polynomial and degree

Sorry if the title is not very explicit, but I didn't find a good title for the problem. In fact, this is the problem : Let $P \in \mathbb{R}[X]$ a polynomial which verify the following condition ...
0
votes
1answer
48 views

Solving a difficult system of equations [closed]

Can anyone help me solve this system of equations: $$ c+d=-a\\ cd=b $$ $$ e+f=-c\\ ef=d $$ $$ a+b=-e\\ab=f $$ (a, b, c, d, e, f are variables).
0
votes
0answers
21 views

Roots of unity and symmetric polynomial

I want to find that $X^n-1 \in \mathbb{Q}[X]$ is a linear combination of $s_1,...,s_{n-1}$ and $s_n +(-1)^n$ with $s_i$ are the elementary symmetric polynomials. I know that $$X^n=\sum \limits_{i=1}^n(...
0
votes
0answers
15 views

Understanding an algebra isomorphism resulting from the fundamental theorem of symmetric polynomials

Let $k$ be a field, and let $s_1, ..., s_n$ be the elementary symmetric polynomials in $k[X_1, ..., X_n]$. The fundamental theorem of symmetric polynomials tells us that Each symmetric polynomial ...
2
votes
2answers
50 views

Higher Algebra by Hall & Knight: Chapter 1, Art 16, Example 3

This is a solved example from Higher Algebra by Hall and Knight. There are a few steps missing so I can't understand the continuity of the solution. Chapter 1, Ratio, Art 16, Example 3 Solve the ...
1
vote
2answers
63 views

Find maximum value

Given $0 \leq a,b,c \leq \dfrac{3}{2}$ satisfying $a+b+c=3$. Find the maximum value of $$N=a^3+b^3+c^3+4abc.$$ I think the equality does not occur when $a=b=c=1$ as usual. I get stuck in finding the ...
0
votes
3answers
89 views

Prove that if $x_1 + x_2 + … + x_n = n$, then $x_1^k + x_2^k + … + x_n^k \ge n$

$x_1, x_2, ..., x_n \in \mathbb R$ are nonegative and $k \in \mathbb R$, $k \ge 1$. Prove that if $x_1 + x_2 + ... + x_n = n$, then $x_1^k + x_2^k + ... + x_n^k \ge n$. I tried to find the smallest ...
-1
votes
1answer
145 views

Show that $\frac {1}{3+x^2+y^2} + \frac {1}{3+y^2+z^2} +\frac{1} {3+x^2+z^2}\leq \frac {3}{5} . $

Let $x, y, z>0$ s.t. $x+y+z=3$. Show that $$\frac {1}{3+x^2+y^2} + \frac {1}{3+y^2+z^2} +\frac{1 } {3+x^2+z^2}\leq \frac {3}{5}\ . $$ My idea: $$3 + x^2 + y^2 \geq 1 + 2x+ 2y=7-2z $$ I notice ...
1
vote
1answer
65 views

If $\alpha$ and $\beta$ are roots of $(x^2)-(4x)-1=0$, find $\sqrt[3]{\alpha}$+ $\sqrt[3]{\beta}$

My question in handwriting https://i.stack.imgur.com/4vPBs.jpg If $\alpha$ and $\beta$ are roots of this equation $$(x^2)-(4x)-1=0$$ Then find $$\sqrt[3]{\alpha}+\sqrt[3]{\beta}$$ Please do not ...
3
votes
2answers
116 views

How to prove $f(x)=x^4$ is concave up by definition?

I know $f(x)=x^4$ is concave up, by calculating its second derivative. However, how to prove that $f(x)=x^4$ is concave up by definition, say $f(\frac{x_1+x_2}{2})<(1/2)f(x_1)+(1/2)f(x_2)$ for all $...
3
votes
3answers
100 views

Prove $\frac{a}{\sqrt{a^2+1}}+\frac{b}{\sqrt{b^2+1}}+\frac{c}{\sqrt{c^2+1}} \le \frac{3}{2}$

If $a.b,c \in \mathbb{R^+}$ and $ab+bc+ca=1$ Then Prove $$S=\frac{a}{\sqrt{a^2+1}}+\frac{b}{\sqrt{b^2+1}}+\frac{c}{\sqrt{c^2+1}} \le \frac{3}{2}$$ My try we have $$S=\sum \frac{a}{\sqrt{a^2+ab+bc+...
0
votes
0answers
24 views

When is the quotient of an algebra by a polynomial algebra a free module?

Let $A$ be a $k$-algebra and let $V$ be a subalgebra of $A$ that is a polynomial algebra. Is $A$ a free module over $V$? One example is when $A=\mathbb{C}[x_1, x_2, \ldots x_n]$ and $V$ is the ring ...
5
votes
4answers
101 views

If $a^2+b^2 \gt a+b$ and $a,b \gt 0$ Prove that $a^3+b^3 \gt a^2+b^2$

I'm not too sure about this, I have been working on for some time and I reached a solution (not really too sure about) Question: If $a^2+b^2 \gt a+b$ and $a,b \gt 0$ Prove that $a^3+b^3 \gt a^2+b^2$ ...
2
votes
1answer
87 views

How to prove inequality $(a^2+b^2+c^2)^3\ge6(a^3+b^3+c^3)^2$ when $a+b+c=0$?

I know the identity $a^3+b^3+c^3-3abc = 1/2 (a+b+c) [(a-b)^2+(b-c)^2+(c-a)^2]$. So when it comes to this problem where $a+b+c=0$, you get $(a^2+b^2+c^2)^3\ge54(abc)^2$ when $a+b+c=0$ (because $ a^3+b^...
2
votes
1answer
97 views

If $\sum\limits_{\text{cyc}}x=\frac\pi2$, then $2\sqrt{\sum\limits_{\text{cyc}}\tan x}\le\sum\limits_{\text{cyc}}\frac{\sqrt{\tan x}}{\cos x}$

Let $x$, $y$, $z$ be positive real numbers such that $x+y+z=\frac{\pi}{2}$. Then, $$2\sqrt{\tan x+\tan y+\tan z}\leq \frac{\sqrt{\tan x}}{\cos x}+\frac{\sqrt{\tan y}}{\cos y} +\frac{\sqrt{\tan z}}{\...
1
vote
7answers
88 views

$x^2 y = 1$, $y^2 z = 128$, $z^2 x = 32$

I tried to do this simultaneous equation in my additional maths class at school, and not even the teacher could do it using elimination method (how we were asked to do it). Through trial and error, ...
0
votes
1answer
25 views

Cyclic vs Symmetric Polynomials

I visited brilliant.org and I found this from the Cyclic Polynomials wiki. Note that for the polynomial shown in the pic, if we leave it in its current form, we might see that, if we change $(x,\ y,\ ...
0
votes
1answer
44 views

Whence Catalan's identity?

An identity attributed to Catalan is: $A^3 = B^2 + C^2 + D^2$ where $ A = x^2 + y^2 + z^2$ $B = x(3z^2 - x^2 - y^2)$ $C = y(3z^2 - x^2 - y^2)$ $D = z(z^2 - 3x^2 - 3y^2) $ . This can be used ...
1
vote
1answer
24 views

Existence of full set of $k$-th roots of unity in $GF(p)$

We all know that if $p$ is prime then for $k = p-1$ in $GF(p)$ (field of integers mod $p$) all non-zero elements of the field constitute the full set of $k$-th roots of unity (Fermat's Little Theorem)....
1
vote
2answers
104 views

How to prove this inequality using AM-GM?

Suppose $a,b,c$ are positive real numbers. Then prove that $$\Big(\frac{a+b}{2}\Big)\Big(\frac{b+c}{2}\Big)\Big(\frac{c+a}{2}\Big)\ge\Big(\frac{a+b+c}{3}\Big)\Big(abc\Big)^\frac{2}{3}\tag{*}$$ My ...
2
votes
2answers
252 views

Prove the inequality using Chebyshev's Inequality

If $a,b,c \in(0,\infty)$, then prove that: $$9(a^3+b^3+c^3)\ge(a+b+c)^3$$ I was trying to prove this inequality using Chebyshev's Inequality and assuming $a\ge b \ge c$ but to no avail. Can please ...
1
vote
1answer
39 views

Show that for positive numbers a,b,c,d, $\sum_{cyc} ab \leq \frac{1}{4}\left(\sum_{cyc} a \right)^2$ and … [duplicate]

Let a,b,c,d be four positive real numbers. Show that $$\sum_{cyc} ab \leq \frac{1}{4}\left(\sum_{cyc} a \right)^2$$ and $$\sum_{cyc} abc \leq \frac{1}{16}\left(\sum_{cyc} a \right)^3$$ My textbook ...
2
votes
5answers
99 views

Find the smallest and highest value of the product $xyz$

Find the smallest and highest value of the product $xyz$ assuming that: $x + y + z = 10$ and $x^2 + y^2 + z^2 = 36$. I calculated this: $x+y+z=10 => (x+y+z)^2=10^2$ $x^2+y^2+z^2+2xy+2yz+2zx=100$...
4
votes
0answers
33 views

Expansion of Elementary Symmetric Functions

Let $$e_k(X_1,X_2,\dots, X_n)=\sum_{1\le i_1<i_2<\dots<i_k\le n}X_{i_1}X_{i_2}\dots X_{i_k}$$ be the elementary symmetric function of degree $k$ and $$p_k(X_1,X_2,\dots, X_n)=\sum_{1\le i\le ...
0
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0answers
17 views

What are the applications of symmetric polynomials and cyclic symmetric polynomials?

I found the procedure to solve them, but I don't understand where they are used in science or other fields. Is the sole reason for their being identified differently to use a procedure to factorize ...
2
votes
0answers
49 views

Elementary Symmetric Polynomials and Determinant

I am trying to show that \begin{equation} \sqrt{\Delta} = \prod\limits_{1 \leq i < j \leq n} \left( x_i - x_j \right) = \det \begin{pmatrix} \dfrac{\partial \sigma_{n,1}}{\partial x_1} & \cdots ...
2
votes
0answers
29 views

Evaluating an elementry symmetric polynomial at some specific points.

Let $n$ be an even positive integer. Let $e_{\frac{n}{2}}(x_1, x_2, \ldots ,x_n)$ denote the ${\frac{n}{2}}^{\textrm{th}}$ elementary symmetric polynomial in $n$ variables $x_1,x_2,\cdots,x_n$, ...
-1
votes
1answer
164 views

Solving 6 equations [closed]

I'm working on one privacy-preserving cryptographic protocol that both parties try to compute one function with their private inputs. If one party is trying to guess the other party's inputs then she ...
0
votes
1answer
389 views

$\;32\displaystyle\left(\sum_{c}\frac{1}{7+(x-3)^2}\right)\leq \sum_{c} \frac{x^2+yz}{y+z}+6$

Let $x, y, z > 0$ Prove that $\;32\displaystyle\left(\displaystyle\sum_{c}\frac{1}{7+(x-3)^2}\right)\leq \displaystyle\sum_{c} \frac{x^2+yz}{y+z}+6$ My work, $\displaystyle\sum_{c} \frac{x^2+...
4
votes
4answers
72 views

Prove $x^n(y-z) + y^n(z-x) + z^n(x-y)$ is divisible by $(y-z)(z-x)(x-y)$

I have to show that $$P(x, y, z) = x^n(y-z) + y^n(z-x) + z^n(x-y)$$ is always divisible by $Q(x, y, z) = (y-z)(z-x)(x-y)$ for $n$ greater than $1$ and I have no idea how to proceed. -> I would ...
3
votes
0answers
100 views

General techniques for proving coefficients of a multinomial are all positive

I have encountered a problem in my research where I need to prove that all the coefficients of a certain multinomial of the form $R_k(\mathbf{x}, \mathbf{y}) = \frac{P_k(\mathbf{x}, \mathbf{y})}{Q_k(\...
0
votes
1answer
37 views

Symmetric Expressions in Quadratic Equation

What is Symmetric Expression in Quadratic Equation? As per the definition of my textbook, - The Symmetric Expressions of the roots $\alpha$, $\beta$ of an equation are those expressions in $\...
1
vote
2answers
105 views

Solving Complex Number Equation with Galois theory

Suppose that the complex numbers $\alpha$, $\beta$ and $\gamma$ satisfy \begin{align*} \alpha + \beta + \gamma &= 3, \\ \alpha^2 + \beta^2 + \gamma^2 &= 5, \\ \alpha^3 + \beta^3 + \gamma^...
1
vote
1answer
160 views

If $x+y+z=3$ then $\sum x\sqrt{x^3+3y} \ge 6$

Let $x,y,z>0$ such that $x+y+z=3$. Prove that $$\sum x\sqrt{x^3+3y} \ge 6$$ This trying doesn't help. With Cauchy Schwarz $(\sum x\sqrt{x^3+3y})^2\geq \sum x^2\sum(x^3+3y) = (x^2 + y^2 + z^2)(x^...
3
votes
2answers
38 views

Analogues of the elementary symmetric polynomials for the alternating group

In the case of three variables, the elementary symmetric polynomials are $$ \begin{align} e_1(X_1,X_2,X_3)&:=X_1+X_2+X_3, \\ e_2(X_1,X_2,X_3)&:=X_1 X_2+X_1 X_3+X_2 X_3, \\ e_3(X_1,X_2,X_3)&...
-3
votes
2answers
300 views

Inequality $\sqrt{xy+yz+zx} \ge \frac {8}{15} (x+y+z)$ [closed]

By Titu's inequality: $\sum_{cyc} \frac {1}{x+y} \ge \frac {(1+1+1)^2}{2(x+y+z)} = \frac {9}{2(x+y+z)}$ Then, to prove: $ \frac {3}{x+y+z} + \sum_{cyc} \frac {1}{x+y} \ge \frac {4}{\sqrt{xy+yz+zx}}$...
4
votes
2answers
228 views

find minimum value of $n$ that satisfies this inequality: $4(x_1^2+x_2^2+…+x_n^2)<2(x_1+x_2+…+x_n)<x_1^3+x_2^3+…+x_n^3$

I saw this question, it looks very hard: there are real positive numbers $\{x_1,x_2,...x_n\}$ , given the inequality: $$4(x_1^2+x_2^2+...+x_n^2)<2(x_1+x_2+...+x_n)<x_1^3+x_2^3+...+x_n^3$$ ...
0
votes
0answers
7 views

Solution to $e_k(X+A)\neq e_k(X)$, where $e_k$ is the $k$th elementary symmetric polynomial

In A survey of linear preserver problems (Pierce et al), pages 31-32, it is said that if $e_k(A)$ is the $k$th elementary symmetric polynomial on the eigenvalues of the matrix $A\in\text{Mat}_n(F)$, $...
-4
votes
3answers
53 views

Find solution following the system of equations! [closed]

$abcde-a=357^{400}$ $abcde-b=359^{410}$ $abcde-c=361^{420}$ $abcde-d=363^{430}$ $abcde-e=365^{440}$ ($a,b,c,d,e$ are natural numbers) I don't have any idea. I just tried this, $$a(bcde-1)=357^{...
-2
votes
1answer
83 views

Inequality about Schur

Let a,b,c be the sides of triangle such that $a+b+c=1$. Prove that $$5(ab+bc+ca)\geq18abc+a+b+c$$ I tried to prove: $$5(ab+bc+ca)\geq18abc+a+b+c$$$$10(ab+ac+bc)\geq36abc+2(a+b+c)$$$$a(5b+5c-2-12bc)+b(...
2
votes
4answers
105 views

How to prove AM-GM by induction 3

Let $a_1;a_2;...;a_n\ge 0$. Prove that $$\frac{\sum ^n_{k=1}a_k}{n}\ge \sqrt[n]{\prod ^n_{k=1}a_k}$$ We will prove it's true with $n=k$. Indeed we need to prove it's true with $n=k+1$ WLOG $a_1\le ...
1
vote
0answers
43 views

Can the “symmetric algebra” over $\mathbb R^n$ be defined from an infinite-dimensional exterior algebra?

https://en.wikipedia.org/wiki/Symmetric_algebra If I understand that article correctly, the symmetric algebra $S(\mathbb R^n)$ is (isomorphic to) the algebra of polynomials with $n$ variables. As a ...
0
votes
1answer
48 views

Symmetric polynomial identities: $(x,y,z)^n$ in terms of $\sigma _1=x+y+z$, $\sigma _2 = xy+yz+xz$ and $\sigma _3 = xyz$

In Arthur Engels "Problem Solving Strategies" book in the section on symmetric polynomials, he asks us to prove the identities below. I read up on expanding trinomials and got the quickest method to ...
2
votes
1answer
144 views

Prove that: $\frac{bc}{a^2+1}+\frac{ac}{b^2+1}+\frac{ab}{c^2+1}\leq \frac{3}{4}$ [duplicate]

Given three positive numbers a,b,c satisfying $$a^2+b^2+c^2=1$$ Prove that: $$\frac{bc}{a^2+1}+\frac{ac}{b^2+1}+\frac{ab}{c^2+1}\leq \frac{3}{4}$$ The things I have done so far: $$\sum \limits_{cyc}\...
0
votes
2answers
79 views

Prove that: $\sum\limits_{cyc} \frac{b+c}{a^{2}+bc}\leq \sum\limits_{cyc} \frac{1}{a}$

If $a$, $b$ and $c$ are positive then $$\sum\limits_{cyc} \frac{b+c}{a^{2}+bc}\leq \sum\limits_{cyc} \frac{1}{a}$$ The things I have done so far: $$\frac{b+c}{a^2+bc}\leq \frac{b+c}{2a\sqrt{bc}}=\...
0
votes
2answers
59 views

If $a$, $b$ and $c$ are positive then $\sum\limits_{cyc} \frac{a^{2}}{b^{2}+c^{2}+bc}\geq 1$

If $a$, $b$ and $c$ are positive then $\sum\limits_{cyc} \frac{a^{2}}{b^{2}+c^{2}+bc}\geq 1$. I tried to solve this problem by C-S. But I can't sovle it. Things I have done so far: $\sum\limits_{cyc}...
1
vote
3answers
104 views

symmetric polynomial recursion to solve the system, $x^5+y^5=33$, $x+y=3$

I was just reading on symmetric polynomials and was given the system of equations$$x^5+y^5=33 \text{ , } x+y=3$$ In the text they said to denote $\sigma_1=x+y$ and $\sigma_2=xy$, and to use recursion....
3
votes
1answer
63 views

Find maximal permutation group $G$ such that a polynomial is $G$-invariant

I don't know if this is a trivial question. But because I lack some background I would need advise or a reference. I have an $n$-variate polynomial over $\mathbb Q$, say $f$, and I am interested in ...
0
votes
1answer
40 views

inequality involving $(xy+yz+zx)^3$ and the $pqr$ method

For non-negative numbers $x$, $y$, and $z$ the claim is that: $$(xy+yz+zx)^3+9 x^2 y^2 z^2 \geq 4 (x+y+z)(xy+yz+zx)xyz$$ Without loss of generality, one may assume that $x+y+z=1$ so that $xyz\le\...