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Questions tagged [symmetric-polynomials]

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

30
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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 &...
-1
votes
0answers
40 views

Question on symmetrical integral polynomials

I am reading a proof of the transcendence of $\pi$ but I am stucked with a problem on polynomials. The proof begins like that: "Suppose ${\beta _1},{\beta _2}, \ldots ,{\beta _m}$ are the roots of an ...
3
votes
2answers
82 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
82 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
41 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
49 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
38 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
91 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
42 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
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0answers
10 views

Pieri /Giambelli /Littlewood-Richardson rules for skew symmetric functions [info]

I am doing my thesis on shifted symmetric functions: theory and computation. I would like to know if anyone could recomend me any pdf or book where I can read about: Pieri rule/formula Giambelli ...
0
votes
0answers
24 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
31 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
77 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
34 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
47 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
45 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
129 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
25 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
54 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
66 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
91 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 ...
0
votes
1answer
152 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
66 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
120 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
105 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
27 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
102 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
88 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
100 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
186 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
35 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
46 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
27 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
105 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
258 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
46 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
104 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
37 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
21 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
54 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
30 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
165 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
468 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
73 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
103 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
39 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
107 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^...