2025/12/13 by Silouanos Brazitikos, Brazitikos, Silouanos, C. Pandis +1 · 1 citation
Mathematics · #Random Matrices and Applications #Geometry and complex manifolds #Mathematical functions and polynomials
paper · pdf · doi:10.48550/arxiv.2512.12254
We prove Hunter's conjecture on complete homogeneous symmetric polynomials. For even n and every integer k≥ 1, we show that under the constraint ∑i=1n ai2=1 the global minimum of the even-degree polynomial h2k(a1,…,an) 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 h2k with the probabilistic representation k! hk(a)=𝔼(∑i=1n aiXi)k, where X1,…,Xn are i.i.d. standard exponential random variables with density e-x1x>0 and a combinatorial identity. This viewpoint further yields sharp upper and lower bounds for 𝔼|∑i=1n aiXi|q under natural constraints on the coefficients, including the spherical constraint ∑ ai2=1 combined with the non-negative regime ai≥0, or the centred regime ∑ ai=0. Moreover, we determine the exact minimum of h2k on the ℓ_∞-sphere S_∞ = \a ∈ ℝn : ‖a‖_∞ = 1\, which yields sharp norm comparison inequalities between the matrix norms induced by complete homogeneous symmetric polynomials and the classical operator and Schatten norms.