Exploring AC Power Flow Resultants through the Lens of Multi-homogeneous Algebraic Geometry

The computational aspects of power systems have been exploring the steady states of AC power flow for several decades. In this paper, we propose a novel approach to AC power flow calculation using resultants and discriminants for polynomials, which are primarily compiled for quadratic power flow equations. In the case of AC power flow nonlinear systems, it is not possible to determine the number of isolated solutions. However, for polynomial systems, the theorem of Bézout is the primary theorem of algebraic geometry. This study considers a certain multi-homogeneous structure in an algebraic geometry system to demonstrate that the theorem of Bézout is indeed a generalization of the fundamental theorem, among other results.