Mathematics Stack Exchange
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Questions tagged [symmetric-polynomials]

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

A symmetric polynomial is a polynomial in several variables which is not changed after any permutation of variables. An important result about symmetric polynomials is the possibility to express any symmetric polynomial using elementary symmetric polynomials. Symmetric polynomials are useful, e.g., in connection with roots of polynomials (Vieta formulae). Another useful result concerns the Newton-Girard formulae, which expresses power sums in terms of elementary symmetric polynomials.

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Is it possible to have three real numbers that have both their sum and product equal to $1$?

I have to solve $ x+y+z=1$ and $xyz=1$ for a set of $(x, y, z)$. Are there any such real numbers? Edit : What if $x+y+z=xyz=r$, $r$ being an arbitrary real number. Will it still be possible to find real $x$, $y$, $z$?
bikalpa
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Rational solutions to $a+b+c=abc=6$

The following appeared in the problems section of the March 2015 issue of the American Mathematical Monthly. Show that there are infinitely many rational triples $(a, b, c)$ such that $a + b + c = abc = 6$. For example, here are two solutions…
user940
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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 electrodynamics, we have Kirchhoffs law and so on, which result…
Nikolaj-K
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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 $f_0, f_1, f_2, \cdots \in \mathbb{Z}[X]$, where $f_0=1$. Now, define…
Krishanu Sankar
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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 & \qquad (3)\\ (a^4+b^4)&(c^4+d^4) &=25 & \qquad (4)\\ \end{split} First I used…
Heart
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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 that there are three distinct, real roots if…
Daniel Trebbien
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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 + x_3 \\ q &=& x_1x_2 + x_1x_3 + x_2x_3 \\ -r &=&…
Matt Gregory
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Algorithm(s) for computing an elementary symmetric polynomial

I've run into an application where I need to compute a bunch of elementary symmetric polynomials. It is trivial to compute a sum or product of quantities, of course, so my concern is with computing the "other" symmetric polynomials. For instance (I…
Juan Joder
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Prove that $\sum\limits_{cyc}\frac{a}{a^{11}+1}\leq\frac{3}{2}$ for $a, b, c > 0$ with $abc = 1$

Let $a$, $b$ and $c$ be positive numbers such that $abc=1$. Prove that: $$\frac{a}{a^{11}+1}+\frac{b}{b^{11}+1}+\frac{c}{c^{11}+1}\leq\frac{3}{2}.$$ I tried homogenization and the BW (https://artofproblemsolving.com/community/c6h522084), but it…
Michael Rozenberg
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A generalization of arithmetic and geometric means using elementary symmetric polynomials

Let $a_1, a_2, \ldots, a_n$ be positive real numbers. A while ago I noticed that if you form the polynomial $$ P(x) = (x - a_1)(x-a_2) \cdots (x-a_n) $$ then: The arithmetic mean of $a_1, \ldots, a_n$ is the positive number $m$ such that $(x -…
Caleb Stanford
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How to prove this inequality? $ab+ac+ad+bc+bd+cd\le a+b+c+d+2abcd$

let $a,b,c,d\ge 0$,and $a^2+b^2+c^2+d^2=3$,prove that $ab+ac+ad+bc+bd+cd\le a+b+c+d+2abcd$ I find this inequality are same as Crux 3059 Problem.
math110
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A generalized (MacLaurin's) average for functions

The average value of a function $y=f(x)$, on an interval $[a,b]$, is ${1\over {b-a}}\int_a^b f(t)dt$. This of course relates to the arithmetic average. It is easy to see that a corresponding formula for the geometric average is $\exp\left({1\over…
Maesumi
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Specializations of elementary symmetric polynomials

Let $\mathcal{S}_{x}=\{x_{1,},x_{2},\ldots x_{n}\}$ be a set of $n$ indeterminates. The $h^{th}$elementary symmetric polynomial is the sum of all monomials with $h$ factors \begin{eqnarray*} e_{h}(\mathcal{S}_{x}) & = & \sum_{1\leqslant…
Félix
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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? $$(a-b)^3 + (b-c)^3 + (c-a)^3 -3(a-b)(b-c)(c-a) = 0$$ Thanks
user42912
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Symmetric functions written in terms of the elementary symmetric polynomials.

[A recent post reminded me of this.] How can we fill in the blanks here: For any _____ function $f(x,y,z)$ of three variables that is symmetric in the three variables, there is a _____ function $\varphi(u,v,w)$ of three variables such that…
GEdgar
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