Differential-Algebraic Extension Field Solutions for Polynomial Equations with Complex Coefficients: A Comprehensive Extension

This paper extends the framework of explicit polynomial solving via differential-algebraic field extensions to equations with complex coefficients. We construct a universal complex differential-algebraic extension field KC 𝑛 for degree 𝑛, generated by abstract symbols representing the critical point, formal derivatives (critical values),roots of unity, and 𝑛th roots of specific rational functions. For any concrete complex polynomial 𝑓 of degree 𝑛,we prove the existence of a specialization homomorphism 𝜄𝑓 : KC 𝑛 → LC𝑓 into a differentially closed extension of its coefficient field that contains explicit representations of all roots.The solution resides in the extended field LC𝑓 , which, for generic polynomials of degree ≥ 5, is not contained in the radical closure Frad, thus extending the admissible solution space while remaining fully consistent with the Abel–Ruffini theorem’s restriction on solutions by radicals. Furthermore, we present the Complex Differential Algebraic Root-Finding (C-DARF) algorithm with O(𝑛2) typical and O(𝑛3) worst-case time complexity,accompanied by comprehensive numerical validation. This work provides a new paradigm for explicit polynomial solving with complex coefficients that is algebraically transparent, numerically stable, and computationally tractable, while opening new research directions at the intersection of differential algebra, Galois theory, and numerical computation.