UC Irvine

This dissertation works towards building a fundamental theory of general $\sigma_k$ equations and general inverse $\sigma_k$ equations, showing up in many different fields. For example, PDE, differential geometry, and complex geometry. Our primary goal is to construct nice algebra tools, especially related to real algebraic geometry, so that we can generalize previous classical equations to more complicated settings. Once the framework is settled, we aim to look for a priori estimates and further obtain the solvability of these equations. To be more precise, first, we introduce a special class of multilinear polynomials and a special class of univariate polynomials which are related to the convexity of these equations. Second, we study a priori estimates of these equations provided that the convexity and a $C$-subsolution are given. Last, by collecting these equations which have a priori estimates, we obtain a special algebraic set to apply the method of continuity and further look for the solvability. As an application, we apply our theory and results to the deformed Hermitian--Yang--Mills equation, an equation discovered around the same time by Mariño--Minasian--Moore--Strominger \cite{marino2000nonlinear} and Leung--Yau--Zaslow using different points of view when studying mirror symmetry in string theory. We confirm one of the conjectures by Collins--Jacob--Yau of the deformed Hermitian--Yang--Mills equation when the complex dimension equals three or four.