Questions tagged [symmetric-polynomials]
Questions on symmetric polynomials, polynomials in several variables that are invariant under permutation of the variables.
6
votes
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
votes
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\...
