On the Convexity of General Inverse $σ_k$ Equations

Abstract

We prove that if a level set of a degree $n$ general inverse $σ_k$ equation $f(λ_1, \cdots, λ_n) = λ_1 \cdots λ_n - \sum_{k = 0}^{n-1} c_k σ_k(λ) = 0$ is contained in $q + Γ_n$ for some $q \in \mathbb{R}^n$, where $c_k$ are real numbers not necessary to be non-negative and $Γ_n$ is the positive orthant, then this level set is convex. As an application, this result justifies the convexity of the level set of all general inverse $σ_k$ type equations, for example, the Monge--Ampère equation, the Hessian equation, the J-equation, the deformed Hermitian--Yang--Mills equation, the special Lagrangian equation, etc. Moreover, we find a numerical condition to verify whether a level set of a general inverse $σ_k$ equation is contained in $q + Γ_n$ for some $q \in \mathbb{R}^n$, which is a way to determine the convexity of this level set.

Figures (3)

• Figure 1: $\alpha_{p}(x)$ of $p(x) = x^5 - 19 \binom{5}{3}x^3 + 64 \binom{5}{2}x^2 - 9 \binom{5}{1}x^1 + 20 \binom{5}{0}x^0$
• Figure 2: Value of $\alpha_{P}$ along the deformation of $(x-5)^2(x^2 + 10x +51)$
• Figure 3: $\lambda_1 \lambda_2 \lambda_3 - 4(\lambda_1 + \lambda_2 + \lambda_3) + 9 = 0$ with $\nu_1 =0.8$, $\tau_1 = 1.9$, and $\mu_1 =3$.

Theorems & Definitions (93)

• Definition 1.1: -stableness
• Theorem 1.1: Convexity of the general inverse $\sigma_k$ equation
• Remark 1.1
• Definition 1.2: Noetherian polynomial
• Theorem 1.2: Positivstellensatz
• Example 1.1
• Definition 1.3: -dominance
• Theorem 1.3: -dominance
• Example 1.2
• Theorem 1.4: Monotonicity of log-concavity ratio