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41 votes
4 answers
5k views

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
23 votes
1 answer
1k views

Symmetric polynomial inequality arising from the fixed-point measure of a random permutation

A somewhat strange elementary polynomial inequality came up recently in my work, and I wonder if anyone has seen other things that are reminiscent of what follows. Given $n$ non-negative reals $a_1, ...
BPN's user avatar
  • 543
17 votes
2 answers
3k views

Maclaurin's inequality on elementary symmetric polynomials of arbitrary real numbers

Is there a universal constant $C$ such that the following statement holds? For concreteness, you may assume $C=10000$. Let $a = (a_1, \ldots, a_n)$ be $n$ arbitrary real numbers. For an integer $k$, ...
nichehole's user avatar
  • 381
11 votes
3 answers
651 views

Polynomial inequality of sixth degree

There is the following problem. Let $a$, $b$ and $c$ be real numbers such that $\prod\limits_{cyc}(a+b)\neq0$ and $k\geq2$ such that $\sum\limits_{cyc}(a^2+kab)\geq0.$ Prove that: $$\sum_{cyc}\...
Michael Rozenberg's user avatar
11 votes
0 answers
615 views

Monotonicity of ratio of symmetric polynomials

The complete homogeneous symmetric polynomials of degree $\ell$ in $n$ variables are defined by \begin{equation*} h_{\ell}(x_1,x_2,\ldots,x_n) = \sum_{1 \leq i_1 \leq i_2 \leq \ldots \leq i_{\ell} \...
Rachid Ait-Haddou's user avatar
11 votes
0 answers
387 views

Inequality for symmetric polynomial functions of log concave variables

Let $(x_i)_{i \ge 1}$ be a log-concave (resp. log-convex) sequence of non-negative real variables. In other words, for $i \ge 2$, we have $x_i^2 \ge x_{i-1}x_{i+1}$ (resp. $x_i^2 \le x_{i-1}x_{i+1}$). ...
René Gy's user avatar
  • 505
6 votes
1 answer
303 views

Representing a symmetric polynomial as a conical sum of squares

This question in inspired by the recent solution to another question. The following inequality for monomial symmetric polynomials in 4 positive variables $x_1,x_2,x_3,x_4$: $$m_{(4, 3, 2, 1)} + m_{(4, ...
Max Alekseyev's user avatar
5 votes
3 answers
509 views

Polynomial inequality $n^2\sum_{i=1}^na_i^3\geq\left(\sum_{i=1}^na_i\right)^3$

Let $n\ge 3$ be an integer. I would like to know if the following property $(P_n)$ holds: for all real numbers $a_i$ such that $\sum\limits_{i=1}^na_i\geq0 $ and $\sum\limits_{1\leq i<j<k\leq n}...
Michael Rozenberg's user avatar
4 votes
1 answer
324 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
  • 6,406
4 votes
0 answers
55 views

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(...
Annemarie Kästner's user avatar
3 votes
1 answer
154 views

Bounds on symmetric polynomials in power-sum form with bounded coefficients

Let $\boldsymbol{x}=(x_1,\ldots,x_n)$ be a real vector. Define the normalized power-sum symmetric polynomials by $\pi_j(\boldsymbol{x})=\frac 1n(x_1^j+\cdots+x_n^j)$. For a partition $\lambda= (j_1,\...
Brendan McKay's user avatar
2 votes
0 answers
159 views

Behavior of elementary symmetric polynomials near zero sets

It is straightforward to show (see Characterizing intersection of zero sets of elementary symmetric polynomials on R^n) that the set of points $\Lambda_{k}$ in $x \in \mathbb{R}^{n}$ with $\sigma_{k}(...
Will's user avatar
  • 21
2 votes
0 answers
228 views

Multivariate polynomial with positive coefficients

This question was originally asked at stack exchange (https://math.stackexchange.com/questions/292922/multivariate-polynomial-with-all-coefficients-positive), but did not receive any feedback for more ...
Ewan Delanoy's user avatar
1 vote
5 answers
774 views

Inequality with symmetric polynomials [closed]

How to prove the inequality $a^6+b^6 \geqslant ab^5+a^5b$ for all $a, b \in \mathbb R$?
Constantor's user avatar