Questions tagged [symmetric-polynomials]

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40 votes
4 answers
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The sum of squared logarithms conjecture

I am searching for the first proof of (or counterexample to) the following conjecture. (The sum of squared logarithms conjecture) For all natural numbers $n$ and positive numbers $x_1,x_2, \ldots , ...
Patrizio Neff's user avatar
5 votes
0 answers
967 views

A generalization of the difference of squares identity

Let us find explicit integer functions for the coefficients of the monomial expansion of $$ Q \left( x_1, \ldots , x_n \right) = \prod_{\left( \kappa_1, \ldots , \kappa_{n-1} \right) \in \{-1,1\}^{n-1}...
PalmTopTigerMO's user avatar
4 votes
1 answer
509 views

Derivatives of Riemann $\xi$ and traces of zeros

Looking for references essentially corroborating (to authoritatively satisfy some editors) the sketch below of the relationship between even power (2,4,...) sums (traces) of the imaginary part of the ...
Tom Copeland's user avatar
  • 9,937
13 votes
2 answers
2k views

Diagonal invariants of the symmetric group on $k[X_1,X_2,...,X_n,Y_1,Y_2,...,Y_n]$

This sounds like something that must have been answered long ago, but for some reason I can find nothing on it in the internet. (There has been lots of recent activity in diagonal covariants, related ...
darij grinberg's user avatar
11 votes
2 answers
2k views

Generalizing the Fundamental Theorem of Symmetric Polynomials

The fundamental theorem of symmetric polynomials tells us that the ring $\mathbb{Z}[x_1,\ldots,x_n]^{S_n}$ of symmetric polynomials in $n$ variables is generated (without relations) by the elementary ...
Owen Biesel's user avatar
  • 2,326
10 votes
1 answer
456 views

Generalization of symmetric functions

A $n$-variable function $f$ is a symmetric function if $$f(x_1,x_2, \ldots, x_n) = f(x_{\sigma(1)}, x_{\sigma(2)}, \ldots, x_{\sigma(n)})$$ for every permutation $\sigma \in S_n$. In particular, if $f$...
MMM's user avatar
  • 245
5 votes
1 answer
432 views

Polynomial defined recursively by a resultant

Cross posting from MSE. Definition: For any natural number $n\ge 3$, define the polynomial $P_{n}\left(x_1,x_2,...,x_{n-1},x_{n} \right)$, with indeterminates $x_{i}$, where $i\in\{1,2,...,n-1,n\}$, ...
PalmTopTigerMO's user avatar
4 votes
1 answer
306 views

Nonnegativity locus of Schur polynomials

Let $a_1,\ldots,a_n \in \mathbb{C}$ be complex numbers that are the zeros of a real polynomial (meaning that the non-real ones come in complex conjugate pairs). Suppose that these numbers are such ...
Tobias Fritz's user avatar
  • 5,775
3 votes
1 answer
308 views

Polynomial function defined recursively by a resultant - is it well defined?

Preliminaries Let $ n $ be an integer such that $ n \geq3 $. Denote $ \left[ n \right] \equiv \{1,2, \ldots ,n \} $. Let $ P $ be a non-empty subset of $ \left[ n \right] $ such that $ \left|P \right| ...
PalmTopTigerMO's user avatar
3 votes
2 answers
782 views

On a positivity property of Hall-Littlewood polynomials

Here's the new, more thought through version. Consider a sequence of nonnegative integers $\lambda=(\lambda_1,\ldots,\lambda_n)$ with $\lambda_i\ge \lambda_{i+1}+2$ (the weight $\lambda-2\rho$ is ...
Igor Makhlin's user avatar
  • 3,493