# Questions tagged [symmetric-polynomials]

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

936 questions
Filter by
Sorted by
Tagged with
47 views

### Prove $(\frac{a}{b-c})^2+(\frac{b}{c-a})^2+(\frac{c}{a-b})^2 \geq 2$ [duplicate]

If $a, b, c$ are distinct real numbers, prove that $(\frac{a}{b-c})^2+(\frac{b}{c-a})^2+(\frac{c}{a-b})^2 \geq 2$ I thought of using AM-GM but that is surely not getting me anywhere ( Maybe some ...
49 views

22 views

### Explicit expression for certain Schur polynomials

I am trying to find explicit expression for Schur polynomials of the form \begin{equation} s_\lambda(1,q,q^2,\dots, q^{d-1},q^{-c} ,q^{d+1}, \dots,q^{N-1})~, \end{equation} with $c\in \mathbb{Z}^+$ ...
57 views

72 views

18 views

109 views

137 views

27 views

### Algorithm to compute complete homogeneous symmetric polynomials

Is there any algorithm to compute complete homogeneous symmetric polynomials efficiently? I was able to find algorithm to compute elementary symmetric polynomials. Example :- a1 = 2, a2 =3 So for ...
80 views

I am attempting to prove the following expression, using elementary row and column operations. I have included the attempted solution. $$\det \begin{bmatrix} b^2+c^2 & ab & ac \\ ba & ... 1answer 74 views ### Prove \frac{3}{2} +\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} \leqq \frac{a}{b}+\frac{b}{c} +\frac{c}{a} For a,\,b,\,c>0. Prove:$$\frac{3}{2} +\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} \leqq \frac{a}{b}+\frac{b}{c} +\frac{c}{a}$$My work: After a lot of caculates, I found: \text{RHS-LHS}= ... 0answers 68 views ### Macdonald's symmetric functions and Hall polynomials, Chapter 2, Lemma (1.7) \mathfrak{o} is a discrete valuation ring (local P.I.D), \mathfrak{p} is the unique maximal ideal. \pi is the generator of \mathfrak{p}, i.e. \mathfrak{p}=<\pi>. The residue field k=\... 3answers 46 views ### If x = b+c-a, y = c+a-b, z = a+b-c, prove x^3+y^3+z^3 - 3xyz = 4(a^3+b^3+c^3 -3abc) I got this problem in a book. While trying to solve this I got something like$$2(a+b+c)(a^2+b^2+c^2)$$and can't move forward. Your help will be appreciated. 3answers 75 views ### Prove that  a^2+b^2+c^2 \le a^3 +b^3 +c^3  If  a,b,c  are three positive real numbers and  abc=1  then prove that a^2+b^2+c^2 \le a^3 +b^3 +c^3  I got a^2+b^2+c^2\ge 3 which can be proved  a^2 +b^2+c^2\ge a+b+c . From here how can I ... 0answers 9 views ### How do you show a symmetric function can be factorized into a product of univariate functions? I have a symmetric function given by$$f(x):=\frac{\det\left[{p+k\choose q+k}^{-1}{}_1F_1\left(\begin{matrix}q+k\\p+k\end{matrix};x_j\right)\right]_{j,k=1}^n}{\det[x_j^{k-1}]_{j,k=1}^n},$$for some ... 3answers 86 views ### show this inequality \sum_{cyc}\frac{1}{5-2xy}\le 1 let x,y,z\ge 0 and such x^2+y^2+z^2=3 show that$$\sum_{cyc}\dfrac{1}{5-2xy}\le 1$$try:$$\sum_{cyc}\dfrac{2xy}{5-2xy}\le 2$$and$$\sum_{cyc}\dfrac{2xy}{5-2xy}\le\sum_{cyc}\dfrac{(x+y)^2}{\frac{...
Preliminaries Let $\mathbb{F}$ be a field such that $\operatorname{char}(\mathbb{F})\neq2$. Let $n$ be a non-zero natural number. Let $\mathbb{F}\left[x_1,x_2,\ldots,x_n \right]$ be a polynomial ...