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Questions tagged [symmetric-polynomials]

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

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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 $...
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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+...
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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 ...
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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 ...
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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 ...
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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+...
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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 $p(x)$: $...
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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 $$...
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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 \...
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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 ...
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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)=\...
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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 ...
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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 $$s_{\mu}s_{\nu}=\sum_{\lambda} C^{\lambda}_{\mu, \nu} s_{\lambda}$$ ...
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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 ...
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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 ...
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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 ...
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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 +...
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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....
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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}{...
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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 ...
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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 ...
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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 ...
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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 &...
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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 ...
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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 ...
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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.
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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}+\...
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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 ...
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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}{(...
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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 ...
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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} ...
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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: ...
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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. \...
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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 ...
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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 ...
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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
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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 +...
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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 $a,b,c$.
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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 ...
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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(...
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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 ...
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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 ...
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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 ...
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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 ...
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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+...
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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 $a^3+b^3 \gt a^2+b^2$ ...