Questions tagged [symmetric-polynomials]
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132 questions
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What are the examples of Hermitian Positive Polynomials that are not r-SOS?
A complex polynomial $\mathbb{C}[x,\bar{x}]$ is called Hermitian if it satisfies $$p(x, \bar{x}) = \overline{p(x, \bar{x})}.$$
This means that the polynomial outputs real values. The polynomial can be ...
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Existence of odd integer solutions to a system of elementary symmetric equations
Consider the following system of equations:
$$
\sigma_{2k}(x_1,x_2,\dots,x_{2n+1}) = (-1)^k \binom{n}{k}, \quad 1 \le k \le n,
$$
where $\sigma_k(x_1,x_2,\dots,x_{2n+1})$ denotes the $k$-th elementary ...
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Regularity of a variant of elementary symmetric polynomials
Let $Y_1, Y_2,\dots, Y_n$ be $n$ bases of linear forms in the polynomial ring $k[x_1,\dots, x_n]$, where $k$ is a field or $\mathbb{Z}$. Examples of bases $Y_i$'s are $A_i\cdot[x_1,\dots, x_n]^T$, ...
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Maximal ideals in rings with polynomial relations
Let $F$ be a perfect field, and let $p(t) \in F[t]$ be an irreducible monic polynomial of degree $n$ such that: $$p(t) = t^n+s_1 t^{n-1}+s_2 t^{n-2}+\dots+s_n$$ Let $\theta_1, \theta_2, \dots, \...
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Is $(1+z_1 + \dotsb +z_1^m)^p \dotsb (1+z_n + \dotsb +z_n^m)^p$ Schur positive for large enough $p$?
I would like to know if the symmetric polynomial $(1+z_1 + \dotsb +z_1^m)^p \dotsb (1+z_n + \dotsb +z_n^m)^p$ is Schur positive for large enough $p$. Here, how large the $p$ is can depend on $n$ and $...
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Expressing factorial elementary symmetric polynomials in terms of factorial complete symmetric polynomials
There are well known identities relating elementary symmetric polynomials in $k$ variables and complete homogeneous symmetric polynomials in the same set of variables. For example,
$$e_{2,k}(x) = h_{1,...
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Inequality of elementary symmetric polynomials written in terms of derivatives of $(s+ta_1)\dotsm(s+ta_n)$
For positive integers $k,n$ with $k\le n{-}1$ and $a\in\mathbb{R}^{n}$, let $S_{k}(a):=\sum_{i_{1}<\dots<i_{k}}a_{i_{1}}\dotsm a_{i_{k}}$ be the $k$-th elementary symmetric polynomial. If the ...
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Positivity of elementary symmetric polynomials under linear fractional transformations
The general question
For $1\leq k\leq n$, let $$e_k(a_1,\dots,a_n):=\sum_{j_1<\dots<j_k}a_{j_1}\cdots a_{j_k}$$ be the $k$-th elementary symmetric polynomial.
Let $a_1,\dots,a_n<1$ and $e_1(...
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Expansion of key polynomials in terms of non-symmetric Hall-Littlewood polynomials and charge-like statistics
Edit: The problem I pose here is impossible to solve with the basis $H$, in the answer I made to this post I explain why. The only way I can think it to amend the situation would be to try with ...
- 161
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Asymptotics of generalized exponents of highest weight modules
Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $H^k$ be the space of homogeneous degree $k$ harmonic polynomials in $\mathrm{Sym}(\mathfrak{g}^*)$ and $H\subset\mathrm{Sym}(\mathfrak{g}^*)...
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Hall-Littlewood polynomials for $n$-tuples that are not partitions
For some calculations related to the unramified principal series of ${\rm GL}(n)$ over a $p$-adic field, I need to compute Hall-Littlewood polynomials that are associated to $n$-tuples that are not ...
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Sum of derivative of polynomial over its simple roots
Let $P$ and $Q$ be polynomials over $\mathbb C$, and $n\in\mathbb N$ be a positive integer. I'm interested in the root sums of the form
$$ \sum_{P(x)=0}\frac{Q(x)}{P'(x)^n},$$
where the sum runs over ...
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Matrix with elementary symmetric polynomials as entries
Let $n\geq 1$, and for each $j=1,\ldots, n+1$ let $\mathbf{X}_{j}=(X_{j1},\ldots, X_{jn})$ be $n$ variables. Let $M$ be the $(n+1)\times (n+1)$ matrix whose $(i,j)$-th entry is $$M_{ij}=(-1)^i e_{i-1}(...
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Proving an identity for flagged Schur without use of determinants?
In proposition 3 of Determinantal transition kernels for some interacting particles on the line, Dieker and Warren prove the following identity: consider vector $a:=(a_1,\dotsc,a_N)$ and kernels
$$\...
- 5,644
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Irrational elements can always be moved
Let $x_1,x_2,x_3,\ldots,x_n$ be the roots of a polynomial $P_n(x)$. Let $F$ be the field $\mathbb{Q}[x_1,x_2,x_3,\ldots,x_n]$, i. e. all the possible combinations of rational numbers with $x$'s.
It's ...
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