Questions tagged [symmetric-polynomials]
Questions on symmetric polynomials, polynomials in several variables that are invariant under permutation of the variables.
1,063
questions
2
votes
1answer
64 views
Prove that $\sqrt{a^2-a+1}+\sqrt{b^2-b+1}+\sqrt{c^2-c+1}\ge a+b+c$ [duplicate]
For $a,b,c>0;abc=1.$ Prove that $$\sqrt{a^2-a+1}+\sqrt{b^2-b+1}+\sqrt{c^2-c+1}\ge a+b+c$$
I will post my solution in the answer. Now I'm looking forward to another solution.
4
votes
2answers
105 views
Prove that $\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\leq 3+\sqrt{3}$
Problem. (Nguyễn Quốc Hưng) Let $0\le a,b,c\le 3;ab+bc+ca=3.$ Prove that $$\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\leq 3+\sqrt{3}$$
I have one solution but ugly, so I 'd like to find another. I will post my ...
5
votes
0answers
198 views
Existence of a factorisation proof of the AM-GM inequality
Consider the following formulation of the AM-GM Inequalities:
$$\large f_n(a_1, a_2, a_3,\cdots,a_n) := \left(\sum_{k=1}^n a_k^n\right) - n \prod_{k=1}^n a_k \geqslant 0$$
As required by the classical ...
0
votes
0answers
19 views
Expressing the discriminant of a cubic polynomial in terms of the elementary symmetric polynomials
In David Cox's book GALOIS THEORY, p 41 Exercise 16 (c), he asks for the cube roots of unity to show that x^3-1 has discriminant -27 which will be the coefficient of one of the terms in the expression ...
1
vote
1answer
42 views
Help with an expression of the Schur polynomial
I am reading: https://www2.math.upenn.edu/~peal/polynomials/schur.htm where it says:
The Schur polynomial is defined as $$s_\lambda(x_1,...,x_n)=\prod_{1\le i<j\le n}(x_i-x_j)^{-1}\det(x_j^{\...
1
vote
1answer
40 views
How to express symmetric polynomials in terms of elementary symmetric polynomials?
Firstly, I know there are very similar questions (How to express a symmetric polynomial in terms of elementary symmetric polynomials and Expressing a symmetric polynomial in terms of elementary ...
0
votes
1answer
22 views
Is a system of elementary system of complex symmetric polynomials solvable in polynomial time?
Give a generalized system of the form...
$$a_1 + a_2 + a_3 = x_1 + x_2 + x_3$$
$$a_1 a_2 + a_1 a_3 + a_2 a_3 = x_1 x_2 + x_1 x_3 + x_2 x_3$$
$$a_1 a_2 a_3 = x_1 x_2 x_3$$
... solvable in polynomial ...
9
votes
0answers
184 views
On the Abstract Concreteness Method (bka $ABC-$Method).
I was reading Zdravko Cvetkovski's excellent book Inequalities: Theorems, Techniques, and selected problems, when I arrived at the $16$th chapter: the $ABC-$Method. I had some questions related to ...
0
votes
1answer
27 views
Closed form coefficients of Elementary Symmetric Polynomial specialized to the natural numbers
I am writing a program that computes a Lagrange polynomial using parallel processing for data points situated at every natural number. But I can have $n$ many data points, which will potentially ...
2
votes
2answers
104 views
If $a^2 + b^2 +c^2 =4$ and $a^3 + b^3 + c^3 = 8$ then find $a^4+b^4+c^4$
This is a popular question, but I can find $a^4+b^4+c^4$ if I know $a+b+c, a^2+b^2+c^2, a^3 +b^3 +c^3$ or in special case $a+b+c=0$ we only need a more condition like $a+b+c$. I have tried in many ...
0
votes
1answer
61 views
Find $\sum_{k=1}^{1010}(a_k^k+2+\frac{1}{a_k^k})^{2020}$
Given a polynomial
$P(x)=x^{2020}+x^{2019}+x^{2018}+...+x^2+x+1$
with roots $a_1,a_2,a_3,...,a_{2020}$ Find the value of
$\sum_{k=1}^{1010}(a_k^k+2+\frac{1}{a_k^k})^{2020}$
we can write to be
$\sum_{...
1
vote
0answers
40 views
Invariant Polynomial in the matrices over $\mathbf{R}$
Let $ A = \mathbf{R}[M_n(\mathbf{R})]$ be the ring of polynomials in the real $n$ by $n$ matrices (which nothing more than the polynomial ring in $n^2$ variables).
We say that $P \in A$ is invariant (...
2
votes
0answers
52 views
efficient way of calculating the $n-1$ polynomial coefficients
Given $n$ polynomials with the same $M$ degree.
$$
A_1 = \sum_{i=0}^M a_{1i} x^i \\
A_2 = \sum_{i=0}^M a_{2i} x^i \\
\cdots \\
A_n =\sum_{i=0}^M a_{ni} x^i
$$
My objective is to obtain all ...
1
vote
0answers
39 views
A problem in polynomials to show there are polynomials $f_{1}(x), \cdots f_{k}(x)$ such that $ P(x) = \sum_{j = 1}^{k} (f_{j} (x))^{2}$ [duplicate]
Let $P(x) \in \mathbb{R}$ be non negative $\forall x \in \mathbb{R}$. Prove That, for some $k$, there are polynomials $f_{1}(x), \cdots f_{k}(x)$ such that $$ P(x) = \sum_{j = 1}^{k} (f_{j} (x))^{2}$$
...
1
vote
1answer
26 views
Finding range of the expression $\sum_{cyc}\frac{|x+y|}{|x|+|y|}$ where $x,y,z\in\mathbb{R}-\{0\}$
I am supposed to find the range of the following. At present I do not know how to get started. Obviously as is the case with numerous symmetrical expressions, one of the extrema occur when all ...
0
votes
0answers
31 views
Integral of the Vandermonde determinant.
For which values of $\omega_1,\dots,\omega_d$ the following integral of the Vandermonde determinant:
$$\mathcal{F}[V](\omega_1,\dots,\omega_d)=\int_\Lambda e^{-i\sum_j \theta_j \omega_j}V(e^{i\theta_1}...
0
votes
0answers
44 views
Higher Algebra. What topic is this and where I can read about it?
I want to find some information about how to solve this problem but I don't even know what topic is this. Maybe it's related to symmetric functions. I've tried searching in books but nothing similar ...
1
vote
2answers
37 views
How many terms are there in a cyclic summation?
I was wondering whether there is any formula that can calculate the number of terms in a cyclic sum:
If $i = 1, \ldots, n$, then for $\sum_{\text{cyc}}x_ix_j$, there are $\frac{n(n-1)}{2}$ terms in ...
1
vote
2answers
64 views
$(ab+bc+ca)^3=abc(a+b+c)^3$, prove that $a,b,c$ are in $G.P.$ [duplicate]
Suppose $a,b,c$ are non-zero real numbers such that
$$(ab+bc+ca)^3=abc.(a+b+c)^3$$
Prove that $a,b,c$ must be terms of a $G.P.$
I simplified this equation too
$$(ab)^3+(bc)^3+(ca)^3=abc.(a^3+b^3+c^3)$...
2
votes
1answer
65 views
Little Laurent Polynomial Identity
I am trying to prove the following statement. I don't have any assumptions on the the base field $\Bbbk$ yet, but I am happy to restrict this if required.
Statement: Let $p(h)\in\Bbbk[h^{\pm 1}]$. If $...
2
votes
0answers
92 views
Algebricaic independence of elementary symmetric polynomials
I have a doubt regarding the proof that symmetric elementary polynomials are algebrically independent. These are
\begin{aligned}s_{0}(x_{1},x_{2},\dots ,x_{n})&=1,\\[10pt]s_{1}(x_{1},x_{2},\dots ,...
2
votes
2answers
64 views
Find maximize of $P=\frac{x\sqrt{yz}}{\sqrt{x^2+1}\sqrt[4]{\left(y^2+4\right)\left(z^2+9\right)}}$
Let $x,y,z\in \mathbb{R^+}$ such that $6x+3y+2z=xyz$. Find maximize of $$P=\frac{x\sqrt{yz}}{\sqrt{x^2+1}\sqrt[4]{\left(y^2+4\right)\left(z^2+9\right)}}$$
We will prove $$P\le \sqrt{\frac{16}{27}}$$
$...
1
vote
1answer
94 views
Prove $3\left(9-5\sqrt{3}\right) \sum \frac{1}{a} \geqslant \sum a^2+\frac32\cdot\frac{\left[(\sqrt3-2)(ab+bc+ca)+abc\right]^2}{abc}$
Let $a,\,b,\,c$ are positive real numbers satisfy $a+b+c=3.$ Prove that
$$3\left(9-5\sqrt{3}\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) \geqslant a^2+b^2+c^2 + \frac32 \cdot \frac{\left[(\...
1
vote
6answers
82 views
If $b^2+c^2+bc=3$ then $b+c\leq 2$
Suppose that $b,c\geq 0$ such that $b^2+c^2+bc=3$. Prove that $b+c\leq 2$.
I tried to do that by contradiction but I failed.
Indeed, if $b+c>2$ then $b^2+2bc+c^2>4$ then $(b^2+bc+c^2)+bc>4$. ...
0
votes
0answers
24 views
Generating representations of U(n) using Schur functions.
We know that Schur functions $S_\lambda$ are related with irreps. of $U(n)$ and that there is an associated branching rule for the subgroup chain $U(1)\subseteq \ldots \subseteq U(n-1)\subseteq U(n)$, ...
1
vote
4answers
80 views
Let $x,y,z$ are the lengths of sides of a triangle such that $x+y+z=2$. Find the range of $xy+yz+xz-xyz$ .
Let $x,y,z$ are the lengths of sides of a triangle such that $x+y+z=2$. Find the range of $xy+yz+xz-xyz$ .
What I Tried: I have tried by doing $(x+y+z)(x+y+z)=2*2 = 4.$
Also I got $2(xy+yz+xz)+(x^2+...
1
vote
1answer
76 views
For $n =3,$ write $\Delta^2$ as an element of $A = \mathbb{Q}[e_{1}, e_{2}, e_{3}.]$(manually)
Here is the question I want to answer letter $(e)$ of it manually:
Let $B = \mathbb{Q}[x_{1}, ... , x_{n}] \cong \mathbb{Q}^{[n]}$ and $A = \mathbb{Q}[e_{1}, ... , e_{n}]$ where $e_{i} \in B$ is the ...
1
vote
2answers
60 views
The square root of the discriminant function
Define the discriminant by $$\Delta^2 = \prod_{i > j}(x_i -x_j)^2$$
Clearly $\Delta^2$ is a symmetric polynomial. My question is, why is the square root of of $\Delta^2$, i.e $$\Delta = \prod_{i &...
2
votes
1answer
77 views
Find the value of $\frac{1}{(pq)^2-10r^2+300}+\frac{1}{(qr)^2-10p^2+300}+\frac{1}{(pr)^2-10q^2+300}$
If $p+q+r=6,pq+pr+qr=8,pqr=2$, what is the value of:
$$\frac{1}{(pq)^2-10r^2+300}+\frac{1}{(qr)^2-10p^2+300}+\frac{1}{(pr)^2-10q^2+300}$$
I tried to change $\frac{1}{(pq)^2-10r^2+300}$ to ...
0
votes
1answer
43 views
Compute elementary symmetric functions via MATLAB
For any given numbers $\{x_1,...,x_n\}$, the k-th elementary symmetric function is
$$
E_k=\sum_{1\le i_1<...<i_k\le n} x_{i_1}x_{i_2}...x_{i_k},~~~~~~~k=1,...,n.
$$
Given a specified n-tuple of ...
5
votes
3answers
139 views
prove that $\sum_{cyc}\frac{a}{b^2+c^2}\ge \frac{4}{5}\sum_{cyc}\frac{1}{b+c}$
prove that $$\sum_{cyc}\frac{a}{b^2+c^2}\ge \frac{4}{5}\sum_{cyc}\frac{1}{b+c}$$ for positives $a,b,c$
Attempt: By C-S; $$\left(\sum_{cyc}\frac{a}{b^2+c^2} \right) \left(\sum_{cyc} a(b^2+c^2) \right)\...
2
votes
2answers
76 views
prove $\sum_{cyc}\frac{a^2}{a+2b^2}\ge 1$
prove $$\sum_{cyc}\frac{a^2}{a+2b^2}\ge 1$$ holds for all positives $a,b,c$ when $\sqrt{a}+\sqrt{b}+\sqrt{c}=3$ or $ab+bc+ca=3$
Background Taking $\sqrt{a}+\sqrt{b}+\sqrt{c}=3$ This was left as an ...
1
vote
1answer
42 views
Inequality about elementary symmetric function
For any given numbers $\{x_1,...,x_n\}$, the k-th elementary symmetric function is
$$
E_k=\sum_{1\le i_1<...<i_k\le n} x_{i_1}x_{i_2}...x_{i_k},~~~~~~~k=1,...,n.
$$
I guess the $E_k$ can be ...
1
vote
0answers
42 views
Schur-Positivity of a simple polynomial
Let $\chi_{d,p;f}$ be the following symmetric polynomial,
$$\chi_{d,p;f}(x)=\prod_{l=1}^d\sum_{k=0}^p x_l^{f_k},$$
where $f=\lbrace f_0,\ldots,f_p\rbrace$ is a set of integers. I need to identify for ...
1
vote
0answers
49 views
Prove that $\Sigma a_\nu = \Sigma b_\nu$ (Remmert, exercise 2, section 1 chapter 0)
I'm studying Complex analysis from Remmert Theory of Complex Functions book and I'm trying to do the exercises. However, I'm stuck with exercise 2, section 1, chapter 0. It says:
Let $a_1,...,a_n, b_1,...
1
vote
0answers
26 views
Express the weighted power sums
Fix $m:=(m_1\ldots,m_k)\in\mathbb{N}^k$ be a sequence of natural numbers and for $i\in\mathbb{N}$ denote by $p_{i,m}:=m_1X_1^i+\dots+ m_k X_k^i$ - a weighted power sum polynomial. It seems that the ...
0
votes
2answers
66 views
Symmetric inequality over 3 variables with restriction $a+b+c=1$
Suppose that $a,b,c$ are positive real numbers with $a+b+c=1$. Prove that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\leq 3+\frac{2(a^3+b^3+c^3)}{abc}.$$
Since I am texting from cellphone I will skip some ...
2
votes
4answers
96 views
Minimize $(x+y)(y+z)(z+x)$ given $xyz(x+y+z) = 1$
$x,y,z$ are positive reals and I am given $xyz(x+y+z) = 1$. Need to minimize $(x+y)(y+z)(z+x)$. Here is my approach.
Using AM-GM inequality
$$ (x+y) \geqslant 2 \sqrt{xy} $$
$$ (y+z) \geqslant 2 \...
0
votes
4answers
114 views
prove that $\sum_{cyc}\frac{a(b^2+c^2)}{a^2+bc}\ge 3$
if $a,b,c$ are positive prove $$\sum_{cyc}\frac{a(b^2+c^2)}{a^2+bc}\ge 3$$ given $ab+bc+ca=3$
My try:using AM-GM and Titu's lemma :
$$\sum_{cyc}\frac{a(b^2+c^2)}{a^2+bc}\ge 2\sum_{cyc}\frac{abc}{a^2+...
2
votes
1answer
98 views
Finding all real $(a,b,c)$ satisfying $a+b+c=\frac1{a}+\frac1{b}+\frac1{c}$ and $a^2+b^2+c^2=\frac1{a^2}+\frac1{b^2}+\frac1{c^2}$
I have been trying to find my error in the following question for a while, but am yet to succeed:
Find all triples $(a,b,c)$ of real numbers that satisfy the system of equations:
$$\begin{align}
a+b+...
2
votes
3answers
59 views
Inequality with a, b, c about finding minimal and maximal value
Find the minimal and maximal value (if they exist) of ${\sqrt{\frac{a(b+c)}{b^2+c^2}}} +{\sqrt{\frac{b(a+c)}{a^2+c^2}}} +{\sqrt{\frac{c(b+a)}{b^2+a^2}}}$ if are non-negative real numbers, such that ...
0
votes
1answer
59 views
Multivariate symmetric polynomial system [closed]
Consider the following system of polynomial equations in $x_1,\ldots,x_6$:
\begin{eqnarray*}
x_1x_2 - α & = & 0\\
x_3x_4 - β & = & 0\\
x_5x_6 - γ & = & 0\\
x_1x_4+x_3x_2 - δ &...
6
votes
5answers
132 views
Proving $6(x^3+y^3+z^3)^2 \leq (x^2+y^2+z^2)^3$, where $x+y+z=0$
Question :
Let $x,y,z$ be reals satisfying $x+y+z=0$. Prove the following inequality:$$6(x^3+y^3+z^3)^2 \leq (x^2+y^2+z^2)^3$$
My Attempts :
It’s obvious that $x,y,z$ are either one negative and ...
1
vote
0answers
52 views
Find $f(a,b,c),g(a,b,c)\geqslant 0$ so that $7abc-\sum ab(a+b)= f ( a, b, c )+ ( \,a- b \, ) \cdot g ( a, b, c)$
A friend asked me$:$
For $a,b,c \in [\,1,2\,].$ Find $f(a,b,c)$ and $g(a,b,c)$ where $f(a,b,c),g(a,b,c)$ are all non-negative polynomials such as$:$ $$f= 7abc- ab\left ( a+ b \right )- bc\left ( b+ c ...
4
votes
3answers
93 views
Proving $(a+b+c) \Big(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\Big) \leqslant 25$
For $a,b,c \in \Big[\dfrac{1}{3},3\Big].$ Prove$:$
$$(a+b+c) \Big(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\Big) \leqslant 25.$$
Assume $a\equiv \text{mid}\{a,b,c\},$ we have$:$
$$25-(a+b+c) \Big(\dfrac{...
5
votes
6answers
149 views
prove $a^3+b^3+c^3+3abc\ge \sum_{cyc}ab(a+b)$
prove $$a^3+b^3+c^3+3abc\ge \sum_{cyc}ab(a+b),$$$a,b,c>0$
Obviously this is a direct consequence of the third degree schur's inequality.
I was wondering if this could be proved without this ...
4
votes
0answers
55 views
Is there a way to convert a constrained double sum to unconstrained sum?
I have a sum that is of the form
$$S_{p}(x,y)=\sum_{n=1}^{p-1}\sum_{m=1}^{p-1-n} A_{n,m}(x,y),$$
where $A_{n,m}(x,y)$ is a monomial of the form $c_{n,m}x^ny^m$.
I wish to take a $p\rightarrow\infty$ ...
2
votes
2answers
146 views
Prove $\sum ab \sum \frac{1}{(a+b)^2} \geqslant \frac{9}{4}+\frac{kabc\sum (a^2-bc)}{(a+b+c)^3(ab+bc+ca)}$ for the best k.
For $a,b,c\geqslant 0;ab+bc+ca>0.$ Find $k_\max$ and proving in that case$:$
$$(ab+bc+ca)\Big(\dfrac{1}{(a+b)^2}+\dfrac{1}{(b+c)^2}+\dfrac{1}{(c+a)^2}\Big) \geqslant \dfrac{9}{4}+\dfrac{kabc(a^2+b^...
4
votes
2answers
114 views
Proving $4\Big(\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2} \Big)+\frac{81}{(a+b+c)^2}\geqslant{\frac {7(a+b+c)}{abc}}$
For $a,b,c>0.$ Prove$:$ $$4\Big(\dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2} \Big)+\dfrac{81}{(a+b+c)^2}\geqslant{\dfrac {7(a+b+c)}{abc}}$$
My proof is using SOS$:$
$${c}^{2}{a}^{2} {b}^{2}\Big( \...
0
votes
5answers
152 views
To prove $(a^2+b^2+c^2)(ab+bc+ca)^2 \ge (a^2+2bc)(b^2+2ca)(c^2+2ab)$.
I think that there is not a simple proof of this inequality. (for all real numbers)
My Attempt $1$:
Equality occurs for $a=b=c$.
It is quit sure that the inequality is homogeneous.
So normalize with ...
