Questions tagged [symmetric-polynomials]
Questions on symmetric polynomials, polynomials in several variables that are invariant under permutation of the variables.
1
vote
1answer
36 views
Sum expansion of the elementary symmetric polynomials
Recently I stumbled upon the following equation:
$$e_k(V\cup W) = \sum_{i=0}^{|W|}e_{k-i}(V)e_i(W)$$
$V$ and $W$ are subsets of $\{x_0,x_1,...,x_n\}$, and $V\cap W = \ \varnothing$. Where both $...
0
votes
2answers
31 views
How to solve underdetermined systems of polynomial equations?
I am trying to solve under determined simultaneous non - linear equations, where the variables are multiplied, but the power of the variables is always 1, is there a formal way doing it?
For eg: $x+y+...
1
vote
2answers
49 views
Decide for which $x,y,z$ the following equation system is met: ${1+x+y=xy}$ $2+y+z=yz$ and $5+z+x=zx$
I need to decide for which $x,y,z$ the following equation system is met:
$$ 1+x+y=xy $$ $$ 2+y+z=yz $$
$$5+z+x=zx$$
I can see that $x=0, y=0, z=0$ is not a solution.
I tried to divide ...
0
votes
0answers
31 views
Characters of orthogonal groups as symmetric functions
The Schur functions are characters of irreps of the unitary group, $s_\lambda(U)={\rm Tr}\left(R_\lambda(U)\right)$. They are symmetric functions of the eigenvalues of $U$ and can be written in terms ...
2
votes
3answers
51 views
Inequalities, but working around an absolute value
So what I want to prove is
$$ |xy+xz+yz- 2(x+y+z) + 3| \leq |x^2+y^2+z^2-2(x+y+z)+3| $$
for $x,y,z\in \mathbb{R}$, and I'm aware that the RHS is just $|(x-1)^2+(y-1)^2+(z-1)^2|$.
Now I'm able to ...
2
votes
2answers
63 views
Inequality with a+b+c=1 and $18(a^4+b^4+c^4)+6(a^2+b^2+c^2)+1\geq24(a^3+b^3+c^3)$
Let $a,b,c$ be reals with $a+b+c=1$. Show that :
$$18(a^4+b^4+c^4)+6(a^2+b^2+c^2)+1\geq24(a^3+b^3+c^3).$$
I have tried to something like this:
$$18a^4-24a^3+6a^2-12a+12\geq 0$$
$$18b^4-24b^3+6b^2-12b+...
1
vote
0answers
14 views
Vieta's Formula for Chebyshev basis
Let $p(x)=x^d+\sum_{i=0}^{d-1} a_ix^i$. Then Vieta's formula tells us that the $a_i$ can be expressed as signed elementary symmetric polynomials of the roots $\{\alpha_1,\ldots,\alpha_d\}$ of $p(x)$: $...
3
votes
2answers
52 views
Find $K=a^2b+b^2c+c^2a$ for roots $a>b>c$ of a cubic.
If $a>b>c$ are the roots of the polynomial $P(x)=x^3-2x^2-x+1$ find the value of $K=a^2b+b^2c+c^2a$.
Using Vièta's formulas:
$$a+b+c=2$$
$$ab+bc+ca=-1$$
$$abc=-1$$
Using those I found that
$$...
2
votes
1answer
77 views
What is the Computational Complexity of the Elementary Symmetric Polynomials
The elementary symmetric polynomials in $n$ variables, $e_k(X_1,\dots,X_n)$, are defined implicitly by
$$(X-X_1)(X-X_2) \cdots (X-X_n)=\sum_{k=0}^n (-1)^k e_k(X_1,\dots,X_n) X^{n-k}, \quad 1 \leq k \...
2
votes
1answer
39 views
Write formula in elementary symmetric polynomials
Consider the expression
$$
\prod_{i\in I, j\in J} (x-\alpha_i - \beta_j)
$$
as polynomial in $x$. As this expression is symmetric in the $\alpha_i$ and in the $\beta_j$, you should be able to write ...
1
vote
1answer
37 views
shifted symmetric polynomials
BACKGROUND
When defining shifted symmetric polynomials we do it in the following way:
Let $\mu=(\mu_1,..., \mu_n)$ be a partition with length less or equal to $n$.
Then
$$s_{\mu}^*(x_1,...,x_n)=\...
1
vote
1answer
93 views
filtered algebra vs graded algebra
BACKGROUND
When reading Okounkov-Olshanski's paper about shifted symmetric functions, they define $\Lambda^*$ to be the algebra of shifted symmetric functions.
They say that $\Lambda_n^*$ is a ...
1
vote
2answers
18 views
skew Schur functions $C^{\lambda}_{\mu, \nu}$
When working with skew Schur functions, they can be defined as follows.
Let $C^{\lambda}_{\mu, \nu}$ be the integers such that
$$s_{\mu}s_{\nu}=\sum_{\lambda} C^{\lambda}_{\mu, \nu} s_{\lambda}$$
...
4
votes
3answers
68 views
$\Lambda = \varprojlim\Lambda_n$ (ring of symmetric functions)
This question is related to this other question.
When understanding how it is defined the ring of symmetric functions, I can not see why is so much important to take the inverse limit in the category ...
1
vote
0answers
39 views
Proofs of Newton's identities
I am reading the article "Newton's Identities" by D. G. Mead. Newton's Identities capture the the relation between the elementary symmetric functions of $x_1, x_2,\ldots,x_n$ and the sums of the ...
1
vote
1answer
41 views
Some interesting systems of equations [closed]
1, Solve the system of equations:$\left\{\begin{matrix}
x^3+y^3+2z^3=19x-11y-5z+1\\ x^3+(y^2+1)x=(x^2+y^2)z+z
\\ \sqrt{2+x^2+y^2-2yz}=y^2+z^2-2xy+\sqrt{2}
\end{matrix}\right.$
2,Solve the system of ...
0
votes
1answer
45 views
Algebra precalculus problem
I need to solve this problem and I don’t know how.
If $y^2 + z^2 + yz = a^2$
$z^2 + x^2 + zx = b^2$
$x^2 + y^2 + xy = c^2$
$yz + zx + xy = 0,$ then
$a \pm b \pm c = 0$
I can see that $a^2 + b^2 +...
0
votes
1answer
69 views
symmetric functions vs symmetric polynomials
I am doing my thesis related with symmetric functions and representations.
It is for this reason that I am reading MacDonald's book Symmetric Functions and Hall Polynomials.
When reading chapter 1....
0
votes
1answer
74 views
Solving an interesting polynomial with degree 4? [duplicate]
So the equation is as follows:
$$ 6x^2 -\ 25x \ + 12 \ +\ \frac6{x^2}\ + \frac{25}{x} = 0$$
So one thing that is immediately observable is that pairs of roots will be of the from $$x_1=-\frac{1}{...
4
votes
1answer
94 views
Relation of complete homogeneous symmetric polynomials and the elementary symmetric polynomials
I was reading about the symmetric polynomials and saw the following relation:
$$\sum _{{i=0}}^{m}(-1)^{i}e_{i}(X_{1},\ldots ,X_{n})h_{{m-i}}(X_{1},\ldots ,X_{n})=0\text{ for } m>0$$
The proof ...
0
votes
0answers
90 views
Symmetric group action on polynomial ring
Let the symmetric group $S_4$ act on $\mathbb R[x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4]$ by permuting the 1st $4$ variables and again permuting the last $4$ variables. We can restrict the action to the ...
0
votes
1answer
37 views
On coefficients of $(t-x_1)(t-x_2)..(t-x_i)$
I was reading Emil Artin's Galois Theory (2nd edition). On pp.39-40, Artin defines a number of things as follows.
Let $k$ be a field and $E=k(x_1,\ldots,x_n)$ be the field of all rational functions ...
32
votes
4answers
2k views
Super hard system of equations
Solve the system of equation for real numbers
\begin{split}
(a+b) &(c+d) &= 1 & \qquad (1)\\
(a^2+b^2)&(c^2+d^2) &= 9 & \qquad (2)\\
(a^3+b^3)&(c^3+d^3) &= 7 &...
3
votes
2answers
85 views
Advice on solving these simultaneous (quadratic/cubic) equations?
I have the following simultaneous equations:
$$a x^2 + (b+2ay)x - c_1 = 0$$
$$ay^2 + (b+2ax)y - c_2 = 0$$
Where I'd like to solve for $x$ and $y$.
Obviously $a,b,c_1,c_2$ are known constants. They ...
4
votes
1answer
92 views
chebyshev's inequality - Question
I had a question in my exam and they asked to prove that
prove that:
$$3(1+a^2+a^4)\geq(1+a+a^2)^2$$ for all $a\in\mathbb R$.
Now , I solved it , but the problem is that in the answer they wrote ...
0
votes
2answers
45 views
How to solve the following system of equations?
$\left\{ \begin{aligned} xy + 2x + 2y &= -8\\ yz + 2y + 2z &= 24\\ xz + 2x + 2z &= -11 \end{aligned} \right.$
I need to solve it over the set of real numbers.
3
votes
3answers
50 views
Solve the system $x^2(y+z)=1$ ,$y^2(z+x)=8$ and $z^2(x+y)=13$
Solve the system of equations in real numbers
\begin{cases} x^2(y+z)=1 \\ y^2(z+x)=8 \\z^2(x+y)=13 \end{cases}
My try:
Equations can be written as:
\begin{cases}\frac{1}{x}=xyz\left(\frac{1}{y}+\...
2
votes
0answers
40 views
Are elementary symmetric polynomials evaluated in powers of double cosines of rational multiples of $\pi$ integer?
For $n,k,m\ge 1$ integer, define
$$S(n,m,k)=\sum_{A\subset\{1,\ldots,n\},\\\#A=k}\left(\prod_{a\in A}(-1)^a2^m\cos^m\left(\frac{a\pi}n\right)\right).$$
In other words, $S(n,m,k)$ is the $k$'th ...
0
votes
1answer
105 views
For $x$, $y$, $z$ the sides of a triangle, show $\sum_{cyc}\frac{yz((y+z)^2-x^2)}{(y^2+z^2)^2}\ge\frac{9(y+z-x)(x+z-y)(x+y-z)}{4xyz}$
in $\triangle ABC$, let $AB=z,BC=x,AC=y$,show that
$$\sum_{cyc}\frac{yz((y+z)^2-x^2)}{(y^2+z^2)^2}\ge\frac{9(y+z-x)(x+z-y)(x+y-z)}{4xyz}$$
by well kown Iran 96 inequality
$$(xy+yz+xz)\left(\frac{1}{(...
0
votes
1answer
46 views
programming/coding and polynomials
I would like to know your opinion about which is the best code
to work with mathematical operations/structures.
I am doing my thesis on symmetric polynomials and I would like
to include a ...
0
votes
0answers
26 views
symmetric polynomials in functions
If I have $N$ variables, $a_i$, $i=1,...,N$, then any symmetric polynomial in these can be expressed as a polynomial in the elementary symmetric polynomials:
$$ e_k = \sum_{i_1<...<i_k} a_{i_1} ...
1
vote
0answers
36 views
“Reverse” Vieta's Formulas
In one of my investigations, I needed to figure out sums of powers of roots of polynomial equations. These are not very hard to figure out. For monic polynomials, the first three sums are:
...
1
vote
1answer
80 views
irreducible polynomial $f$ in four variables with complex coefficients such that $f(x^3+y^3+z^3,x^2y+y^2z+z^2x,xy^2+yz^2+zx^2,xyz)=0$
Let $p,q,r,s \in \mathbb C[x,y,z]$ be defined as
\begin{eqnarray*}
p(x,y,z)&=&x^3+y^3+z^3,\\
q(x,y,z)&=&x^2y+y^2z+z^2x,\\
r(x,y,z)&=&xy^2+yz^2+zx^2,\\
s(x,y,z)&=&xyz.
\...
0
votes
1answer
39 views
Writing polynomial as a product of elementary symmetric polynomials
Write $x^2y+xy^2+x^2z+xz^2+y^2z+yz^2 $ as a product of elementary symmetric polynomial
I get $E1=x+y+z$, $E2=xy+xz+yz$, $E3=xyz$.
I've tried factoring out E3(xyz) but I can tell that's not right. I ...
0
votes
1answer
52 views
Graded Ring Category vs Ring Category
I know that in Ring Category we have:
-Objects: Rings.
-Arrows: Ring homomorphisms.
I do not know which are the objects and arrows in Graded Ring Category.
In general, which is the definition of ...
0
votes
3answers
46 views
How do you solve these 2 equations?
$$xy = 1/6$$
$$y+x = 5xy$$
I tried solving them using all methods - substitution, elimination and graphing - but can't get the solutions
0
votes
1answer
35 views
Solving this system equations?
I have to solve this system of equations with $(x,y,z) ∈ ℝ$
$x^2 + y + z = q$
$x+ y^2 + z = q$
$x + y + z^2 = q$
for $q = -1$
So we have:
$x^2 + y + z = -1$ (1)
$x+ y^2 + z = -1$ (2)
$x + y +...
5
votes
1answer
138 views
Solve the system of $3$ quadratic equations [closed]
Consider the system of equation
$${{x}^{2}}+{{(1-y)}^{2}}=a\\
{{y}^{2}}+{{(1-z)}^{2}}=b\\
{{z}^{2}}+{{(1-x)}^{2}}=c$$
Compute $x(1-x)$ in terms of $a,b,c$.
6
votes
0answers
49 views
Symmetric polynomial and degree
Sorry if the title is not very explicit, but I didn't find a good title for the problem. In fact, this is the problem :
Let $P \in \mathbb{R}[X]$ a polynomial which verify the following
condition ...
0
votes
0answers
36 views
Roots of unity and symmetric polynomial
I want to find that $X^n-1 \in \mathbb{Q}[X]$ is a linear combination of $s_1,...,s_{n-1}$ and $s_n +(-1)^n$ with $s_i$ are the elementary symmetric polynomials. I know that $$X^n=\sum \limits_{i=1}^n(...
0
votes
0answers
19 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
62 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
67 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
92 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
vote
2answers
164 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 ...
2
votes
1answer
71 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
126 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
108 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
31 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
104 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 $a^3+b^3 \gt a^2+b^2$
...
