0
votes
1answer
46 views

Proving Newton's identities

Assume $F$ is a field of zero characteristic. Denote the elementary symmetric polynomials of $n$ variables by $e_k$, $\quad k=\overline{1,n}$. Let the symbol $\sum ax_1^{i_1}\dots x_n^{i_n}$ denote ...
1
vote
1answer
65 views

Derivation and application of Newton's identity

How is the following identity derived? $$\sum_{\ell =0}^{n-1}(-1)^\ell e_\ell s_{n-\ell}+(-1)^nne_n=0$$ Is there an example demonstrating the context in which this might be applied?
0
votes
1answer
41 views

Using elementary polynomials to solve system of linear polynomials

Problem Statement I am given a finite set of monic polynomials in t, parameterized by $r_i$ $X_i = t - r_i$ where the $r_i$ are guaranteed unique. Neither $t$ nor $r_i$ are known, only $X_i$. I ...
2
votes
1answer
34 views

If $f$ is an anti-symmetric polynomial, then $f=g\prod_{1\leq i < j\leq n}(X_i-X_j)$ for some $g$ symmetric

So we have the situation that $f\in K[X_1,...,X_n]$ is anti-symmetric, which means that $\sigma (f)=\pm f$ where it is a plus if $\sigma$ is an even permutation on the $X_i$ and a minus if it is not ...
1
vote
0answers
40 views

Non-symmetric polynomials, game

This is a game I thought was easy but appears to be too hard for me... I'm trying to find a polynomial in x,y,z (they commute) such that permutations of the variables only give rise to 2 different ...
1
vote
0answers
57 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 ...
5
votes
7answers
314 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? $$ \left(x^2+\frac{1}{x^2}\right)\left(x^3+\frac{1}{x^3}\right)=23\cdot 110. $$ ...
0
votes
0answers
54 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
votes
1answer
175 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 + ...
0
votes
0answers
36 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 ...
4
votes
2answers
106 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 ...
1
vote
1answer
100 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 $\alpha_i$ ...
1
vote
0answers
49 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 ...
1
vote
2answers
48 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$\sqrt{3}$ ...
1
vote
0answers
73 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}, ...
0
votes
1answer
299 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 ...
1
vote
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
votes
1answer
51 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 ...
1
vote
0answers
40 views

Are the coefficients of $\text{minpoly}(\alpha + \beta)$ polynomials in the coefficients of $\text{minpoly}(\alpha)$ and $\text{minpoly}(\beta)$?

Suppose you are given an algebraic field extension $L \supset K$ and $\alpha,\beta \in L$ with $f(X) = \text{minpoly}_K(\alpha)(X)=a_0+...+a_{m-1} X^{m-1}+X^m$ and ...
1
vote
1answer
42 views

$p(x)$ be a polynomial over $\mathbb{Z}$. If $P(a)=P(b)=P(c)=-1$ with integers $a,b,c$.Then $P(x)$ has no integral roots

Let $\mathbb{P}(x)$ be a polynomial over $\mathbb{Z}$. If $\mathbb{P}(a)$=$\mathbb{P}(b)$=$\mathbb{P}(c)$=$-1$ with integers $a,b,c.$ Then $\mathbb{P}(x)$ has no integral roots
2
votes
0answers
288 views

general biquadratic equation

I'm not sure if I understand what the following question is asking: Show that the solution of the general biquadratic equation $x^{4}+ax^{3}+bx^{2}+cx+d=0$ can be obtained directly, that is, ...
3
votes
3answers
136 views

Ring of invariants of Klein Four group

Assume $F$ is a field and assume $f\in F[x_1,\ldots,x_4]$ is a polynomial that is invariant under the Klein Four group $V_4$. How can I show that this polynomial can then be rewritten as a polynomial ...
2
votes
1answer
135 views

Symmetric polynomial optimization

Recently I asked a stupid question here (there’s no harm in that, even Fields medalist Terence Tao advises to ask dumb questions once in a while). Here is a variant question that may be more ...
1
vote
2answers
157 views

Three inequalities with sums of fractions over two positive integers

In a proof, I arrive at three inequalities for all $p,q \geqslant 0$: \begin{align} \frac{p+1}{q+1} + \frac{q+1}{p+1} &\geqslant 1 + \frac{p}{2q+1} + \frac{q}{2p+1} + \frac{1}{p+q+1};\cr ...
0
votes
0answers
120 views

Showing that an alternating polynomial is the product of some symmetric polynomial and the Vandermonde polynomial

For simplicity, consider polynomials of two variables. Let $f(x,y)$ be an arbitrary alternating polynomial. I want to show that $f(x, y)$ is the product of some symmetric polynomial and the ...
6
votes
2answers
531 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 ...
5
votes
4answers
1k views

Expressing a symmetric polynomial in terms of elementary symmetric polynomials using computer?

Are there any computer algebra systems with the functionality to allow me to enter in an explicit symmetric polynomial and have it return that polynomial in terms of the elementary symmetric ...
11
votes
1answer
2k views

Using Vieta's theorem for cubic equations to derive the cubic discriminant

Background: Vieta's Theorem for cubic equations says that if a cubic equation $x^3 + px^2 + qx + r = 0$ has three different roots $x_1, x_2, x_3$, then $$\begin{eqnarray*} -p &=& x_1 + x_2 ...
3
votes
3answers
209 views

Do these special power functions generate all homogeneous symmetric polynomials?

Over rational numbers, the set of all power functions up to a certain degree generate all symmetric polynomials in that degree. My question is as follows. To be succinct, let's say we have four ...
4
votes
2answers
2k 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 ...
6
votes
3answers
692 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 ...
6
votes
2answers
282 views

Number of terms in a monomial symmetric polynomial

Is there a closed form expression for the number of terms in a monomial symmetric polynomial in a given number of variables for a particular partition of exponents, in terms of which/how many ...
5
votes
2answers
1k views

symmetric polynomials and the Newton identities

I want to write $P(x,y,z)=yx^{3}+zx^{3}+xy^{3}+zy^{3}+xz^{3}+yz^{3}$ in terms of elementary symmetric polynomials, but I'm getting stuck at the first step. I know I should follow the proof of the ...
15
votes
2answers
2k views

Why does the discriminant of a cubic polynomial being less than 0 indicate complex roots?

The discriminant $\Delta = 18abcd - 4b^3d + b^2 c^2 - 4ac^3 - 27a^2d^2$ of the cubic polynomial $ax^3 + bx^2 + cx+ d$ indicates not only if there are repeated roots when $\Delta$ vanishes, but also ...