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 avoid catastrophic cancellation near repeated roots.· A quartic solver based on the quadratic Tschirnhaus transformation, where the auxiliary cubic is solved by our stable cubic solver. Special handling is included for biquadratic cases, zero resolvent roots, and near-degenerate configurations. Extensive numerical experiments on 1000 random complex cubics and 1000 random complex quartics demonstrate that the proposed methods achieve accuracy comparable to state-of-the-art libraries (Jenkins–Traub, Durand–Kerner, NumPy) while being deterministic, iteration-free, and algebraically transparent. A backward stability theorem is provided under the IEEE 754 double-precision model. Ten additional fully worked examples (purely imaginary coefficients, near-double roots, degenerate resolvent, biquadratic cases) are included. Source code in Python and C++ is available in the supplementary material. Keywords: complex cubic equations, complex quartic equations, principal branch, numerical stability, branch cut, Chebyshev polynomials, algebraic solution.