Some Extremal Symmetric Inequalities

06/10/2022

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by   Tetsuya Ando, et al.

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Let ℋ_n,d^s := ℝ[x_1,…, x_n]_d^𝔖_n be the set of symmetric polynomials of degree d, and A be a compact semialgebraic subset of ℙ_ℝ^n-1. We shall study some PSD cones ℙ(A, ℋ_n,d^s) := {f ∈ℋ_n,d^s | f(a) ≥ 0 (∀ a ∈ A)}. In this article, we study extremal elements of 𝒫(ℝ_+^3, ℋ_3,5^s+), 𝒫(ℝ_+^3, ℋ_3,6^s0), and 𝒫(ℝ^4, ℋ_4,4^s) where 𝒫(ℝ_+^3, ℋ_3,6^s0) := {f ∈𝒫(ℝ_+^3, ℋ_3,6^s) | f(1,…,1)=0 }.