Sharp inequalities for symmetric polynomials, Hunter's conjecture, and moments of exponential random variables

TL;DR

This paper proves Hunter's conjecture on symmetric polynomials, establishes sharp bounds for moments of sums of exponential variables, and derives norm inequalities, combining algebraic, probabilistic, and combinatorial methods.

Contribution

It confirms Hunter's conjecture, provides explicit minimizers and closed-form values for symmetric polynomials, and introduces new bounds for exponential moments and matrix norms.

Findings

Proved Hunter's conjecture for even-degree symmetric polynomials.

Derived exact minimizers and closed-form minimal values.

Established sharp bounds for exponential sum moments and matrix norms.

Abstract

We prove Hunter's conjecture on complete homogeneous symmetric polynomials. For even $n$ and every integer $k\geq 1$, we show that under the constraint $\sum_{i=1}^n a_i^2=1$ the global minimum of the even-degree polynomial $h_{2k}(a_1,\dots,a_n)$ is attained precisely at the half-plus/half-minus vector and we compute the optimal value in closed form. The proof combines algebraic properties of $h_{2k}$ with the probabilistic representation $k!\,h_k(a)=\mathbb{E}(\sum_{i=1}^n a_iX_i)^k$, where $X_1,\dots,X_n$ are i.i.d. standard exponential random variables with density $e^{-x}1_{x>0}$ and a combinatorial identity. This viewpoint further yields sharp upper and lower bounds for $\mathbb{E}|\sum_{i=1}^n a_iX_i|^{q}$ under natural constraints on the coefficients, including the spherical constraint $\sum a_i^2=1$ combined with the non-negative regime $a_i\ge0$, or the centred regime $\sum a_i=0$. Moreover, we determine the exact minimum of $h_{2k}$ on the $\ell_\infty$-sphere $S_\infty = \{a \in \mathbb{R}^n : \|a\|_\infty = 1\}$, which yields sharp norm comparison inequalities between the matrix norms induced by complete homogeneous symmetric polynomials and the classical operator and Schatten norms.