Questions tagged [symmetric-polynomials]
Questions on symmetric polynomials, polynomials in several variables that are invariant under permutation of the variables.
936
questions
0
votes
2answers
47 views
Prove $(\frac{a}{b-c})^2+(\frac{b}{c-a})^2+(\frac{c}{a-b})^2 \geq 2$ [duplicate]
If $a, b, c$ are distinct real numbers, prove that
$(\frac{a}{b-c})^2+(\frac{b}{c-a})^2+(\frac{c}{a-b})^2 \geq 2$
I thought of using AM-GM but that is surely not getting me anywhere ( Maybe some ...
3
votes
2answers
49 views
Prove$:$ $\sum\limits_{cyc} (\frac{a}{b+c}-\frac{1}{2}) \geqq (\sum\limits_{cyc} ab)\Big[\sum\limits_{cyc} \frac{1}{(a+b)^2}\Big]-\frac{9}{4}$
For $a,b,c$ are reals and $a+b+c>0, ab+bc+ca>0, (a+b)(b+c)(c+a)>0.$ Prove$:$
$$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} -\frac{3}{2} \geqq (\sum\limits_{cyc} ab)\Big[\sum\limits_{cyc} \frac{...
5
votes
3answers
125 views
Prove $(a^2+b^2+c^2)^3 \geqq 9(a^3+b^3+c^3)$
For $a,b,c>0; abc=1.$ Prove$:$ $$(a^2+b^2+c^2)^3 \geqq 9(a^3+b^3+c^3)$$
My proof by SOS is ugly and hard if without computer$:$
$$\left( {a}^{2}+{b}^{2}+{c}^{2} \right) ^{3}-9\,abc \left( {a}^{3}+...
0
votes
0answers
22 views
Explicit expression for certain Schur polynomials
I am trying to find explicit expression for Schur polynomials of the form
\begin{equation}
s_\lambda(1,q,q^2,\dots, q^{d-1},q^{-c} ,q^{d+1}, \dots,q^{N-1})~,
\end{equation}
with $c\in \mathbb{Z}^+$ ...
2
votes
2answers
57 views
Inequality question.
Let $a,b,c>0$ with $a+b+c=1$. Show that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} \leq 3 + 2\cdot\frac{(a^3 + b^3 + c^3)}{abc}$.
Ohhhkk. So first off,
$\begin{align} a^3 + b^3+ c^3 & =a^3 + b^...
4
votes
0answers
32 views
Using symmetric polynomials to find the discriminant of $x^4 + px + q$ over $\mathbb{Q}$
I'm trying to prove that the discriminant of $x^4 + px + q$ over $\mathbb{Q}$ is $-27p^4 + 256q^3$, where we define the discriminant to be
$$
\Delta_f = \prod_{i < j}(\alpha_i - \alpha_j)^2
$$
I ...
1
vote
1answer
48 views
$\min\left(\sum_{\text{sym}}\frac{(2x^2+y)(4x + y^2)}{\underset{\ne0}{(2x + y - 2)^2}} - 3(\underset{\gt0}{x}+\underset{\gt0}{y})\right)=?$
Given positives $x$ and $y$ such that $2x + y \ne 2$ and $x + 2y \ne 2$, calculate the minimum value of $$\frac{(2x^2 + y)(4x + y^2)}{(2x + y - 2)^2} + \frac{(2y^2 + x)(4y + x^2)}{(2y + x - 2)^2} - 3(...
1
vote
0answers
9 views
Symmetric functions for multidimensional variables
I have $N$ variables (let's call them $X$) of dimensionality $D$, that I want to symmetrize.
For $D=1$ I know I can use e.g. the elementary symmetric polynomials to accomplish this. What if $D>1$...
1
vote
3answers
37 views
Writing explicitly $(s^2-1)^2+(t^2-1)^2$ as a polynomial in $st$ and $s+t$?
Consider the symmetric polynomial
$$ P(s,t)=(s^2-1)^2+(t^2-1)^2.$$
How can we write $P$ as a polynomial in the variables $st,t+s$?
The Fundamental theorem of symmetric polynomials implies this is ...
1
vote
2answers
59 views
Proving inequality by SOS.
For $x,y,z>0.$ Prove$:$ $$P={x}^{4}y+{x}^{4}z+3\,{x}^{3}{y}^{2}-11\,{x}^{3}yz+3\,{x}^{3}{z}^{2}+3
\,{x}^{2}{y}^{3}+3\,{x}^{2}{y}^{2}z+3\,{x}^{2}y{z}^{2}+3\,{x}^{2}{z}^{
3}+x{y}^{4}-11\,x{y}^{3}z+3\,...
1
vote
1answer
49 views
Find the stronger inequality of $\frac{1}{ab+2c^{2}+2c}+\frac{1}{bc+2a^{2}+2a}+\frac{1}{ca+2b^{2}+2b}\geq \frac{1}{\sum ab}$
For $a,b,c>0$ and $a+b+c=1.$ Prove$:$ $$\frac{1}{ab+2c^{2}+2c}+\frac{1}{bc+2a^{2}+2a}+\frac{1}{ca+2b^{2}+2b}\geqq \frac{1}{ab+bc+ca}$$
This inequality is easy and there are two nice proof by AM-GM ...
1
vote
1answer
69 views
Prove Newton's Identities using the properties of a symmetric polynomial.
Apologies if I didn't explain this properly in the title. I understand how we progress from (1) down, but I don't understand how to use the coefficients of EQ 5 to prove Newton's Identities.
...
7
votes
2answers
91 views
Prove $\frac{x^2+yz}{\sqrt{2x^2(y+z)}}+\frac{y^2+zx}{\sqrt{2y^2(z+x)}}+\frac{z^2+xy}{\sqrt{2z^2(x+y)}}\geqq 1$
For $x,y,z>0$ and $\sqrt{x} +\sqrt{y} +\sqrt{z} =1.$ Prove that$:$
$$\frac{x^2+yz}{\sqrt{2x^2(y+z)}}+\frac{y^2+zx}{\sqrt{2y^2(z+x)}}+\frac{z^2+xy}{\sqrt{2z^2(x+y)}}\geq 1$$
My solution$:$
Let $x=...
2
votes
3answers
72 views
Prove $(a+b+c)^3 (a+b-c)(b+c-a)(c+a-b) \leqq 27a^2 b^ 2 c^2$
For $a,b,c>0$$,$ prove$:$ $$(a+b+c)^3 (a+b-c)(b+c-a)(c+a-b) \leqq 27a^2 b^ 2 c^2$$
My proof by S-S method$,$ see here.
Another proof by $pqr$ method$:$
Let $p=a+b+c,\,q=ab+bc+ca,\, r=abc.$ This ...
5
votes
4answers
122 views
Find minimal value of $\left(2-x\right)\left(2-y\right)\left(2-z\right)$
Let $x,y,z>0$ such that $x^2+y^2+z^2=3$. Find minimal value of $$\left(2-x\right)\left(2-y\right)\left(2-z\right)$$
I thought the equality occurs at $x = y = z = 1$ (then it is easy), but the fact ...
4
votes
2answers
101 views
Prove the following inequality $\sum_{i<j<k}\frac{a_ia_ja_k}{(n-2)(n-1)n}\le \bigg(\sum_{i<j}\frac{a_ia_j}{(n-1)n}\bigg)^2+\frac{1}{12}$
For positive integer $n \ge 3$, prove the following inequality $$\sum_{i<j<k}\frac{a_ia_ja_k}{(n-2)(n-1)n}\le \bigg(\sum_{i<j}\frac{a_ia_j}{(n-1)n}\bigg)^2+\frac{1}{12}$$ where $a_1+a_2+\...
1
vote
0answers
11 views
Combinatorics for exterior power for arbitrary Specht module
The exterior powers of the standard representation are easily seen to be the representations whose Young diagrams have only boxes in the first row or first column. But, what if I start with an ...
3
votes
4answers
93 views
Prove $\frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b} +\frac{81abc}{4(a+b+c)^2} \geqq \frac{7}{4} (a+b+c)$
For $a,b,c>0$. Prove that$:$
$$\frac{ab}{c}+\frac{bc}{a}+\frac{ca}{b} +\frac{81abc}{4(a+b+c)^2} \geqq \frac{7}{4} (a+b+c)$$
My proof:
We have$:$ $$\text{LHS}-\text{RHS} =\frac{g(a,b,c)}{4abc(a+b+...
0
votes
0answers
18 views
Proof of polynomial kernel is a kernel
Given that $k_1,\ldots,k_n: \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}$ are kernels. Let $c_1, \dots, c_n \in \mathbb{R}^+$ and $p \in \mathbb{N}$.
We have to prove in the following function $k$ ...
3
votes
3answers
61 views
Proving $(a+b+c)^2\prod_{cyc}(a+b)-4\sum_{cyc}(a^2b+a^2c)\sum_{cyc}ab\geqq 0$
From Mr. Michael Rozenberg solution:
For $a,b,c>0$$,$ prove that$:$
$$(a+b+c)^2\prod_{cyc}(a+b)\geq4\sum_{cyc}(a^2b+a^2c)\sum_{cyc}ab,$$
I found two SOS proof:
1) $$\text{LHS-RHS}={\frac { \left( ...
1
vote
0answers
18 views
How to take the inner product of a Schur Function and Skew-Schur function quickly
Consider the Hall-inner product of a Schur Function with a Skew-Schur function. Say for instance $\langle s_{(5,3,2,1)/(3,2,1)} , s_{(3,2)} \rangle$. What is a fast way to compute this?
Since $\...
0
votes
0answers
9 views
Complete homogeneous symmetric functions
I'm getting trouble to understand a definition and I hope you could help.
We define $m_\lambda(x)=\sum_{a}x^a$,with $a$ varying in the set of rearrangements of the length $n$ vector with $x=(x_1,x_2,.....
4
votes
4answers
109 views
Polynomial $x^3-2x^2-3x-4=0$
Let $\alpha,\beta,\gamma$ be three distinct roots of the polynomial $x^3-2x^2-3x-4=0$. Then find $$\frac{\alpha^6-\beta^6}{\alpha-\beta}+\frac{\beta^6-\gamma^6}{\beta-\gamma}+\frac{\gamma^6-\alpha^6}{\...
1
vote
1answer
46 views
If $a,b,c>0$ and $a+b+c=1$, then prove that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\le 3+2\cdot\frac{a^3+b^3+c^3}{abc}$.
Question: If $a,b,c>0$ and $a+b+c=1$, then prove that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\le 3+2\cdot\frac{a^3+b^3+c^3}{abc}$.
Solution: Observe that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\le 3+2\...
2
votes
1answer
137 views
How do I show that $\sum_{\text{cyc}}\frac{3a^2-2ab+3b^2}{(a+b)^2} < \frac{9}{4}(\frac{a}{c} + \frac{c}{a}) - \frac{3}{2}$?
Let the real numbers $a,b,c \in \mathbb(0, \infty\ )$, $a\leq b\leq c$. Prove that $$\frac{3a^2-2ab+3b^2}{(a+b)^2}+\frac{3b^2-2bc+3c^2}{(b+c)^2}+ \frac{3c^2-2ac+3a^2}{(c+a)^2} \leq\frac{9}{4}(\frac{a}{...
0
votes
2answers
99 views
Prove $a^2+b^2+c^2 \geqslant \frac{9abc}{a+b+c}+2(1+\sqrt 2)(a-b)(b-c)$
Let $a,b$ and $c$ are positive real numbers. Prove that
$$a^2+b^2+c^2 \geqslant \frac{9abc}{a+b+c}+2(1+\sqrt 2)(a-b)(b-c).$$
My proof is not nice.
Indeed, we need to prove the inequality where $(a−...
0
votes
2answers
51 views
Relation between the roots and coefficient.
Let Let a, b and c be the roots of the equation $$x^3 +3x^2-1=0$$Then what is the value of expression $a^2b+b^2c+c^2a$.
I got it done by evaluate the sum and difference of
$a^2b+b^2c+c^2a$ and $ab^...
2
votes
1answer
55 views
Schur inequality
Show that for all positive real numbers $a$, $b$ and $c$ such that $abc=1$, the inequality $a+b+c+2a^4+2b^4+2c^4\ge \dfrac{3}{2}\left(a^2\left(\dfrac{1}{b}+\dfrac{1}{c}\right)+b^2\left(\dfrac{1}{a}+\...
2
votes
2answers
70 views
Prove $\frac{a^2}{(b+c)^2}+\frac{b^2}{(c+a)^2}+\frac{c^2}{(a+b)^2}+\frac{1}{4}\ge \frac{a^2+b^2+c^2}{ab+bc+ca}$
For $a,b,c>0$. Prove that: $$\frac{a^2}{(b+c)^2}+\frac{b^2}{(c+a)^2}+\frac{c^2}{(a+b)^2}+\frac{1}{4}\geqq \frac{a^2+b^2+c^2}{ab+bc+ca}$$
NguyenHuyen gave the following expression$:$
$$\sum \...
0
votes
1answer
27 views
Algorithm to compute complete homogeneous symmetric polynomials
Is there any algorithm to compute complete homogeneous symmetric polynomials efficiently? I was able to find algorithm to compute elementary symmetric polynomials.
Example :- a1 = 2, a2 =3
So for ...
1
vote
1answer
80 views
Proving $\left|\begin{smallmatrix} b^2+c^2 & ab & ac \\ ba & c^2 +a^2 & bc\\ ca & cb & a^2+b^2\end{smallmatrix}\right|=4a^2b^2c^2$
I am attempting to prove the following expression, using elementary row and column operations. I have included the attempted solution.
$$\det \begin{bmatrix} b^2+c^2 & ab & ac \\ ba & ...
0
votes
1answer
74 views
Prove $\frac{3}{2} +\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} \leqq \frac{a}{b}+\frac{b}{c} +\frac{c}{a}$
For $a,\,b,\,c>0$. Prove: $$\frac{3}{2} +\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} \leqq \frac{a}{b}+\frac{b}{c} +\frac{c}{a}$$
My work:
After a lot of caculates, I found:
$\text{RHS-LHS}=$
...
2
votes
0answers
68 views
Macdonald's symmetric functions and Hall polynomials, Chapter 2, Lemma (1.7)
$\mathfrak{o}$ is a discrete valuation ring (local P.I.D), $\mathfrak{p}$ is the unique maximal ideal. $\pi$ is the generator of $\mathfrak{p}$, i.e. $\mathfrak{p}=<\pi>$. The residue field $k=\...
0
votes
3answers
46 views
If $x = b+c-a$, $y = c+a-b$, $z = a+b-c$, prove $x^3+y^3+z^3 - 3xyz = 4(a^3+b^3+c^3 -3abc)$
I got this problem in a book. While trying to solve this I got something like
$$2(a+b+c)(a^2+b^2+c^2)$$
and can't move forward.
Your help will be appreciated.
5
votes
3answers
75 views
Prove that $ a^2+b^2+c^2 \le a^3 +b^3 +c^3 $
If $ a,b,c $ are three positive real numbers and $ abc=1 $ then prove that $a^2+b^2+c^2 \le a^3 +b^3 +c^3 $
I got $a^2+b^2+c^2\ge 3$ which can be proved $ a^2 +b^2+c^2\ge a+b+c $. From here how can I ...
0
votes
0answers
9 views
How do you show a symmetric function can be factorized into a product of univariate functions?
I have a symmetric function given by
$$f(x):=\frac{\det\left[{p+k\choose q+k}^{-1}{}_1F_1\left(\begin{matrix}q+k\\p+k\end{matrix};x_j\right)\right]_{j,k=1}^n}{\det[x_j^{k-1}]_{j,k=1}^n},$$
for some ...
3
votes
3answers
86 views
show this inequality $\sum_{cyc}\frac{1}{5-2xy}\le 1$
let $x,y,z\ge 0$ and such $x^2+y^2+z^2=3$ show that
$$\sum_{cyc}\dfrac{1}{5-2xy}\le 1$$
try:
$$\sum_{cyc}\dfrac{2xy}{5-2xy}\le 2$$
and
$$\sum_{cyc}\dfrac{2xy}{5-2xy}\le\sum_{cyc}\dfrac{(x+y)^2}{\frac{...
3
votes
1answer
137 views
A certain composition into the elementary symmetric polynomials
Preliminaries
Let $\mathbb{F}$ be a field such that $\operatorname{char}(\mathbb{F})\neq2$.
Let $n$ be a non-zero natural number. Let $\mathbb{F}\left[x_1,x_2,\ldots,x_n \right]$ be a polynomial ...
1
vote
3answers
72 views
Prove $\Big[\sum\limits_{cyc} a(a^2+2bc)\Big]^3 \geqq 3(ab+bc+ca)^2 . \sum\limits_{cyc} a(a^2+2bc)^2$
For $a,b,c>0$, prove that: $$ \Big[\sum\limits_{cyc} a(a^2+2bc)\Big]^3 \geqq 3(ab+bc+ca)^2 . \sum\limits_{cyc} a(a^2+2bc)^2$$
BW works here, but it's very ugly!
My try: Let $p=a+b+c,q=ab+bc+ca,r=...
1
vote
1answer
59 views
Stuck when transforming and solving this
Given abc=1 ( all positive real numbers). Prove that:
$$\frac ab + \frac bc + \frac ca +3( \frac ba +\frac cb +\frac ac) \ge 2(a +b +c+\frac 1a+ \frac 1b +\frac1c)$$
My attempt:
$$\frac ab + \frac ...
4
votes
3answers
126 views
Prove that for all positives $a, b$ and $c$, $(\sum_{cyc}\frac{c + a}{b})^2 \ge 4(\sum_{cyc}ca)(\sum_{cyc}\frac{1}{b^2})$.
Prove that for all positives $a, b$ and $c$, $$\left(\frac{b + c}{a} + \frac{c + a}{b} + \frac{a + b}{c}\right)^2 \ge 4(bc + ca + ab) \cdot \left(\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}\right)$$
...
3
votes
3answers
124 views
Lagrange Method of Solving Cubic Equations
Let $K$ be a field (for simplicity, one may assume $K\subseteq\mathbb{C}$), and let $d\in K$. Denote $w=\sqrt[3]{d}$, and $\zeta=(-1+\sqrt{-3})/2$. If $f(x)=x^3+ax^2+bx+c$ is the minimal polynomial of ...
0
votes
0answers
21 views
A Problem on Symmetric Polynomials.
Consider the biquadratic equation $x^4+4px^3+6qx^2+4rx+s=0$. If $\alpha_{1}, \alpha_{2}, \alpha_{3}$ and $\alpha_{4}$ are the roots of this equation find—
$\sum_{\text {symmetric}}\left(\alpha_{1}-\...
1
vote
1answer
56 views
How to analyze the equation $(x-y)^2=2\big( (x+y)-2\sqrt{xy} \big)$?
Suppose that $x,y$ are positive real numbers and that
$$ (x-y)^2=2\big( (x+y)-2\sqrt{xy} \big). \tag{*}$$
Then Mathematica claims that one of the following $3$ options holds:
$$1. \, \, \, x=y.$$
$...
0
votes
0answers
17 views
What role do symmetries play in solving a quadratic and higher degree polynomial?
I'm writing an essay on Radicals and one of my questions is on symmetries and I was hoping someone would have a bit of insight or could maybe help me reword the question so I undertsand what is being ...
1
vote
1answer
70 views
Prove that $3x^3-41x+48\leq 0$ for $x \in [\sqrt 3, \sqrt 6]$
Prove that $3x^3-41x+48\leq 0$ for $x \in [\sqrt 3, \sqrt 6]$.
This is from an inequality in one of Titu Andreescu’s inequality books. More exactly, $2(a+b+c)\geq 3+\frac38(a+b)(b+c)(c+a)$ for ...
3
votes
1answer
42 views
Are there simple expressions for skew Schur polynomials corresponding to hook-shaped diagrams?
I am a physicist who has been studying (skew) Schur polynomials recently. I am particularly interested in skew Schur polynomials $s_{\lambda/\mu}$, where $\lambda$ is a representation corresponding to ...
3
votes
0answers
40 views
Symmetric polynomials; $a_1,a_2,\ldots,a_n\in\Bbb{C}$ s.t. $\prod_{i=0}^n\left(a_i^k+1\right)=1,\;\forall k\in\Bbb{N}\implies a_1=\dots =a_n=0$
Let $a_1,a_2,\ldots,a_n\in\Bbb{C}$ s.t. $\prod_\limits{i=0}^n\left(a_i^k+1\right)=1,\;\forall k\in\Bbb{N}$.
Prove that $a_1=a_2=\dots=a_n=0$.
I have tried to show that all symmetric elementary ...
2
votes
1answer
63 views
Expansion of $(x+y)^n+(x+z)^n+(y+z)^n-x^n-y^n-z^n$ in terms of elementary symmetric polynomials
Consider the symmetric polynomial in $3$ variables
$$
f_n(x,y,z)=(x+y)^n + (x+z)^n+(y+z)^n - x^n-y^n-z^n
$$
where $n\geq 0$ is an integer. I'm inquiring if there is a closed formula for the ...
1
vote
1answer
60 views
For $a,b,c\ge0$, show $a^2 + b^2 + c^2 = 3$ implies $(a^3 + b^3 + c^3)^2 \geq 3 + 2(a^4 + b^4 + c^4)$.
$a,b,c$ are nonnegative real numbers. If $a^2 + b^2 + c^2 = 3$,
prove that $$(a^3 + b^3 + c^3)^2 \geq 3 + 2(a^4 + b^4 + c^4).$$
I tried using $(a^3 + b^3 + c^3)^2 >= 3(a^3 + b^3 + c^3)abc$ to make ...
