A Unified Numerically Stable Framework for Solving Cubic and Quartic Equations with Complex Coefficients Using Principal Branches, Branch Cut Analysis, and Edge‑Case Handling

This paper extends our previously developed numerically stable solvers for cubic and quartic equations from the real to the complex domain. The core novelty is the systematic use of principal branches for square roots, cube roots, and fourth roots, combined with explicit branch cut analysis (negative real axis), which eliminates ambiguity and ensures backward stability for arbitrary complex coefficients. We provide complete algorithms: · A stabilized cubic solver using the identity UV = -p/3 to