Arithmetic Circuit Complexity and Lower Bounds for the Permanent via Valiant's Algebraic Model

This monograph explores the foundational structures of algebraic complexity theory, specifically focusing on Valiant's algebraic model of computation. We rigorously examine the dichotomy between the complexity classes VP and VNP, characterizing the arithmetic circuit complexity of the Permanent polynomial relative to the Determinant. Central to this analysis is the study of lower bounds for arithmetic circuits, utilizing techniques such as the partial derivative measure introduced by Nisan and Wigderson, and the shifted partial derivative measure developed by Kayal et al. We further discuss the "chasm" at depth three and four, demonstrating how strong lower bounds for constant-depth circuits imply super-polynomial lower bounds for general arithmetic circuits. Finally, we briefly address the geometric complexity theory (GCT) program initiated by Mulmuley and Sohoni, which seeks to resolve the VP vs. VNP conjecture through representation-theoretic obstructions in the orbit closures of the determinant and permanent.