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

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16 views

size of orbit of a polynomial under the action of $S_n$

Given a polynomial $\Phi$ in the $n$ variables $X_1,\ldots,X_n$, the values of $\Phi$ are defined to be the polynomials in the orbit of $\Phi$ under the action of the full symmetric group $S_n$. For ...
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2answers
40 views

Constructing expressions

Suppose you are given all the elementary symmetric functions of $n$ variables $x_1,x_2,...,x_n$ and two rational functions $A(x_1,x_2,...,x_n)$ and $B(x_1,x_2,...,x_n)$ in the same $n$ variables that ...
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0answers
26 views

Unique “splitting fields” for infinite polynomials

When creating the splitting field of a polynomial we can without loss of generality assume it has constant coefficient $1$ and splits as $\prod_{k=1}^n(1-z_kT)$. The splitting field is obtained by ...
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0answers
29 views

Express symmetric polynomial $\prod_{i < j} (X_i+X_j)$ in terms of elementary symmetric functions

Exercise: Define a polynomial $\Sigma(X_1,\ldots,X_n)$ as \begin{align*} \Sigma(X_1,\ldots,X_n) = \prod_{i < j} (X_i+X_j) \end{align*} This is a symmetric polynomial, quite clearly. I want to ...
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3answers
44 views

Elementary Symmetric Polynomials, Roots of cubic polynomials

I'm given $a_1, a_2, a_3$ as roots of the equation $x^3 + 7x^2 - 8x + 3$ and need to find the cubic polynomials with roots $a_1^2, a_2^2, a_3^3$ and $\frac{1}{a_1}, \frac{1}{a_2}, \frac{1}{a_3}$. ...
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1answer
197 views

Understanding the Fundamental Theorem of Symmetric Polynomials within the context of proving $\pi$ transcendental

I am currently studying the proof of the transcendence of $\pi$. There are a bunch of proofs scattered across the web (here, here, and here, to list some); some derive from the Lindemann-Weierstrass ...
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7answers
293 views

If $x+\dfrac{1}{x}=5$, find the value of $x^5+\dfrac{1}{x^5}$.

If $x>0$ and $x+\dfrac{1}{x}=5$, find the value of $x^5+\dfrac{1}{x^5}$. Is there some other way to do find it? $$ \left(x^2+\frac{1}{x^2}\right)\left(x^3+\frac{1}{x^3}\right)=23\cdot 110. $$ ...
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0answers
20 views

Identity with symmetric rational functions

I am trying to prove this identity between rational functions involving symmetrization among variables. Let us consider a set of variables $\{p_1,\ldots,p_n\}$, which I indicate globally as ...
2
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3answers
138 views
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2answers
68 views

Uniqueness of a solution of the system of equations

A friend asked me the following question several days ago, and we still do not have a solution. Prove that the system of equations below has only the solution $(x, y, z)=(1, 1, 1)$. $$ \begin{cases} ...
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2answers
48 views

Have I done something wrong in solving the following pair of equations?

Question given: Solve,$3x^2-5y^2-7=0\\3xy-4y^2-2=0$ What I have done so far: $$ ...
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4answers
105 views

How to solve the following system of equations.

$2x^2-3xy+2y^2=2\frac{3}{4}\\x^2-4xy+y^2+\frac{1}{2}=0$ I tried all the methods that I know, but I could't isolate $x$ or $y$ to form one equation.
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1answer
59 views

Symmetric polynomial and vieta's formulas

Can you help me solve this problem: Express the following symmetric rational function $$\frac{x_1}{-6x_2^2 - 7x_3x_2 - 6x_3^2} + \frac{x_3}{-6x_1^2 - 7x_2x_1 - 6x_2^2} + ...
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2answers
67 views

Number of real solutions for the following set of equations? [closed]

How to solve the following set of equations for real values of $x,y$ and $z$? $$x^2-y^2=z$$ $$y^2-z^2=x$$ $$z^2-x^2=y$$ $(0,0,0)$ is an obvious one solution.
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0answers
33 views

Using Polya's Theorem to check positivity of a multivariate polynomial

I wish to check if a homogeneous polynomial of total degree 4 is positive definite. The polynomial is of the form $$P(u,v,x,y) = \sum_i\alpha_iu^{i_1}v^{i_2}x^{i_3}y^{i_4}$$ with $0 \le i_j \le 2$, ...
3
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1answer
65 views

Factoring $(a+b)(a+c)(b+c)=(a+b+c)(ab+bc+ca)-abc$

How to prove the following equality? $$(a+b)(a+c)(b+c)=(a+b+c)(ab+bc+ca)-abc$$ I did it $$\begin{aligned} a^2b + a^2c + ab^2 + cb^2 + bc^2 + ac^2 + 2abc &=a^2(b + c) + bc(b + c) + a(b^2 + ...
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0answers
25 views

Elementary symmetric polynomials: Why is this sufficient?

Let $x_1,\dotsc,x_n$ be algebraically independent over $K$. $S_n$ acts on $K(x_1,\dotsc,x_n)$ by permuting the $x_i$s, giving rise to distinct field automorphisms of $K(x_1,\dotsc,x_n)$, i.e. a ...
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2answers
83 views

Are the elementary symmetric polynomials “unique”?

The elementary symmetric polynomials are interesting in that they generate the set of symmetric polynomials, in the sense that every symmetric polynomial is some polynomial applied to the elementary ...
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0answers
38 views

Independent indeterminate roots and coefficients of a polynomial

Dummit and Foote, Section 14.6: "If the roots of a polynomial $f(x)$ are independent indeterminates over a field $F$, then so are the coefficients of $f(x)$." This is meant to complete the converse of ...
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2answers
521 views

What is the function space generated by addition and $(a,b)\mapsto (a+b)^{-1}\cdot a\cdot b$ of elements and their inverses?

(the motivation section turned out a little long, the mathematical question is at the end) I need to work with electrical circuts at the moment, computing effective impedances etc. From ...
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2answers
501 views

Number of distinct $f(x_1,x_2,x_3,\ldots,x_n)$ under permutation of the input

$\alpha _n ^n-1=0$ $\alpha _n=e^{2 \pi i/n}$ $$f(x_1,x_2,x_3,\ldots,x_n)=(x_1+\alpha _n x_2+ \alpha _n ^2 x_3+\cdots+\alpha _n ^{n-1} x_n)^n$$ I have read in Jim Brown's paper on page 5 that ...
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0answers
104 views

The close form expression of a Pfaffian

Recall Schur's Pfaffian identity: $$ \mathrm{Pf}\left(\frac{x_j-x_i}{x_j+x_i}\right)_{1\le i,j\le 2n} = \prod_{1\le i<j \le 2n}\frac{x_j-x_i}{x_j+x_i}. $$ Here $x_1,x_2\cdots x_{2n}$ are $2n$ ...
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3answers
104 views

Find all $x,y,z$ satisfying $xy=z-x-y$ and cyclic permutations

Find all ordered pairs $(x,y,z)$ real numbers, which satisfy the following system of equations: $$xy=z-x-y\\xz=y-x-z\\yz=x-y-z$$
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3answers
128 views

How to solve system equation $\left\{\begin{matrix}x^3+y^3+z^3=x+y+z&\\x^2+y^2+z^2=xyz&\end{matrix}\right.$ , $x,y,z\in\mathbb{R}$ ?

How to solve system equation $\left\{\begin{matrix}x^3+y^3+z^3=x+y+z&\\x^2+y^2+z^2=xyz&\end{matrix}\right.$ , $x,y,z\in\mathbb{R}$ ?
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1answer
63 views

symmetric polynomial inequality?

I put $n\ge 2$ balls of various sizes into an urn. I draw two balls (without replacement) from the urn. With each draw, I draw any given ball with probability proportional to its size. Can you ...
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1answer
88 views

Discriminant is zero iff $f\in K[X]$ has repeated roots

I have to prove the statement in the title. Proving from right to left is easy. Now from left to right: $D=(\alpha_1-\alpha_2)^2(\alpha_1-\alpha_3)^2\cdots(\alpha_{n-1}-\alpha_n)^2$ where $\alpha_i$ ...
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3answers
613 views

Three-variable system of simultaneous equations

$x + y + z = 4$ $x^2 + y^2 + z^2 = 4$ $x^3 + y^3 + z^3 = 4$ Any ideas on how to solve for $(x,y,z)$ satisfying the three simultaneous equations, provided there can be both real and complex ...
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2answers
1k views

Sum of cubed roots

I need to calculate the sums $$x_1^3 + x_2^3 + x_3^3$$ and $$x_1^4 + x_2^4 + x_3^4$$ where $x_1, x_2, x_3$ are the roots of $$x^3+2x^2+3x+4=0$$ using Viete's formulas. I know that ...
4
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3answers
69 views

Finding value of equation without solving for a quadratic equation

How do I go about solving this problem: If $α$ and $β$ are the roots of $x^2+2x-3=0$, without solving the equation, find the values of $α^6 +β^6$. In my thoughts: I commenced by expanding $(α ...
7
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0answers
188 views

Basis for $\Bbb Z[x_1,\cdots,x_n]$ over $\Bbb Z[e_1,\cdots,e_n]$

I'm reading the introductory bits in Procesi's Lie Groups, and on p. 22 we have (paraphrasing) Theorem 2. $\mathcal{B}=\{x_1^{\large h_1}\cdots x_n^{\large h_n}: 0\le h_k\le n-k\}$ is a basis for ...
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1answer
39 views

Can we represent a symmetric curve by a parameter with symmetry?

Question : Can we represent the following curve $C$ by one parameter $t$ as $x=f(t),y=g(t),z=h(t)$ with symmetry? The curve $C$ in the $xyz$ space is defined as $$\begin{cases} x^2+y^2+z^2=1 ...
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11answers
1k views

How to prove $(a-b)^3 + (b-c)^3 + (c-a)^3 -3(a-b)(b-c)(c-a) = 0$ without calculations

I read somewhere that I can prove this identity below with abstract algebra in a simpler and faster way without any calculations, is that true or am I wrong? $$(a-b)^3 + (b-c)^3 + (c-a)^3 ...
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2answers
46 views

If $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ are roots of $x^4 +(2-\sqrt{3})x^2 +2+\sqrt{3}=0$ …

Problem : If $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ are roots of $x^4 +(2-\sqrt{3})x^2 +2+\sqrt{3}=0$ then the value of $(1-\alpha_1)(1-\alpha_2)(1-\alpha_3)(1-\alpha_4)$ is (a) 2$\sqrt{3}$ ...
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0answers
87 views

Symmetric function theorem and Galois Theory — How deep is the connection?

By symmetric function theorem in the title, the fundamental theorem of symmetric polynomials is meant: Any symmetric polynomial has a unique representation as a polynomial in the elementary symmetric ...
5
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1answer
78 views

Taking a power of a polynomial to make it symmetric

Suppose I have a non-symmetric multi-variable polynomial in $n$ variables $P(x_1,x_2,...,x_n)$. For example $P$ might be $x_1^2+x_2$ or $x_1-x_2$ Under what conditions will some power $m$ of $P$ ...
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1answer
190 views

Explain the terms : homogeneous , symmetric , anti-symmetric , cyclic with respect to polynomials.

In answers to some of my previous questions , a lot of people used the terms homogeneous polynomial ( in a,b,c ) (under permutations of variables ) , cyclic polynomial ( in a,b,c) (under permutations ...
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0answers
69 views

How to solve this system equation of polynomials?

I have: $F(x) + G(x) = 1 + F(x)*M,$ $G(x) = T_{1}(x) + T_{2}(x) + ... + T_{N}(x)$ $F(x )x^{a_{i}} = T_{i}(x) \times C_{i}(x) + \sum_{j \leq N} T_{j}(x) \times P_{ji}(x)$ In which $M, N, a_{i}, ...
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1answer
146 views

Polynomials invariant under the action of $S_m \times S_n$

The polynomial ring $\mathbb{C}[x_1,\ldots,x_n]$ has a maximal subring invariant under the action of $S_n$ on the variables. This is the ring of symmetric polynomials. Suppose we have ...
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2answers
142 views

A simple 2 grade equations system

If we have: $$x^2 + xy + y^2 = 25 $$ $$x^2 + xz + z^2 = 49 $$ $$y^2 + yz + z^2 = 64 $$ How do we calculate $$x + y + z$$
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1answer
50 views

Factoring the group action on $\prod X_k^k $ gives another group?

Let's say you have a monomial symmetric polynomial, like the following $$ m_{(1,2,3,4)}(X_1,X_2,\dots,X_{10})=X_1^1X_2^2X_3^3X_4^4 +X_2^1X_1^2X_3^3X_4^4 + \text{all permutations...} $$ Then you can ...
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1answer
153 views

An identity on symmetric polynomial

In the polynomial algebra $\mathbb{C}[X_1, X_2,\ldots, X_n]$, we define a set of symmetric polynomials as follows $h_i(X_k, X_{k+1}, \ldots, X_n)$ = sum of all monomials of total degree $i$ in the set ...
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0answers
157 views

Symmetric polynomials

I've got a seemingly simple question that I've become curious about as a result of supervising some undergraduate research. Let's suppose we have some sequence of polynomials $f_0, f_1, f_2, \cdots ...
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0answers
40 views

Clarification on proof including symmetric polynomials

This is regarding theorem 3 in this article. My problems begin after the equalities denoted by (5). My problems aren't so much about theory really, I think. I'm disregarding the authors' notation a ...
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0answers
69 views

Is there a subgroup of $S_{10}$ having $5040$ elements other than $S_7$?

I'm trying to generate to monomial symmetric polynomial $$ m_{(1,2,3,4)}(X_1,X_2,\dots,X_{10})=X_1^1X_2^2X_3^3X_4^4 + \text{all permutations,} $$ starting from the first element by applying all ...
2
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0answers
70 views

are elementary symmetric polynomials concave on probability distributions?

Let $S_{n,k}=\sum_{S\subset[n],|S|=k}\prod_{i\in S} x_i$ be the elementary symmetric polynomial of degree $k$ on $n$ variables. Consider this polynomial as a function, in particular a function on ...
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0answers
36 views

Some lemma on elementary symmetric polynomial

$$e_t(x_2,\ldots,x_n) = \sum^{n-t}_{i=1} (-1)^{i+1} \frac{e_{t+i}(x_1,\ldots,x_n)}{x^i_1}\text{ for every } 0\leq t < n.$$ $e_t$ is the $t^{th}$ elementary symmetric polynomial in the variabel ...
2
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0answers
44 views

Expansions of symmetric polynomials in terms of Jack symmetric polynomials

I was wondering if someone could help me with some Jack polynomial calculations. (I use the notation of I.G. Macdonald's book "Symmetric Functions and Hall Polynomials") Those of you familiar with ...
4
votes
2answers
322 views

Need help solving a particular system of non-linear equations analytically

How would one go about analytically solving a system of non-linear equations of the form: $a + b + c = 4$ $a^2 + b^2 + c^2 = 6$ $a^3 + b^3 + c^3 = 10$ Thanks!
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0answers
26 views

signature of pseudo-Riemannian metric made of Newton polynomials

Given a polynomial with roots $x_1,\ldots,x_n$ and real coefficients, it can be written $$ P(x)=\prod_{i=1}^n \left( x-x_i \right);$$ define Newton polynomials $$s_k(x_1,\ldots,x_n):=\sum_{i=1}^n ...
0
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1answer
44 views

How to prove this symmetric polynomial equations?

I got a problem from a friend, which is to prove that $\Sigma _{i=1}^{n}% \frac{x_{i}^{m}}{\Pi _{j\neq i}(x_{i}-x_{j})}=0$ for m < n-1. I tried to multiply the left of equation with $\Pi _{1\leq ...