[Paper Review] The Fundamental Theorem of Algebra made effective: an elementary real-algebraic proof via Sturm chains

Michael Eisermann|arXiv (Cornell University)|Aug 1, 2008

Polynomial and algebraic computation18 references3 citations

TL;DR

This paper presents a constructive, real-algebraic proof of the Fundamental Theorem of Algebra using Sturm chains and an algebraic formulation of Gauss's winding number. By formalizing the winding number in real-closed fields and leveraging Sturm's algorithm for root counting, the authors provide an elementary, algorithmic proof that every nonconstant complex polynomial has a complex root, with immediate translation into a computable root-finding procedure.

ABSTRACT

Sturm's theorem (1829/35) provides an elegant algorithm to count and locate the real roots of any real polynomial. In his residue calculus (1831/37) Cauchy extended Sturm's method to count and locate the complex roots of any complex polynomial. For holomorphic functions Cauchy's index is based on contour integration, but in the special case of polynomials it can effectively be calculated via Sturm chains using euclidean division as in the real case. In this way we provide an algebraic proof of Cauchy's theorem for polynomials over any real closed field. As our main tool, we formalize Gauss' geometric notion of winding number (1799) in the real-algebraic setting, from which we derive a real-algebraic proof of the Fundamental Theorem of Algebra. The proof is elementary inasmuch as it uses only the intermediate value theorem and arithmetic of real polynomials. It can thus be formulated in the first-order language of real closed fields. Moreover, the proof is constructive and immediately translates to an algebraic root-finding algorithm.

Motivation & Objective

• To provide a constructive, elementary proof of the Fundamental Theorem of Algebra using only real algebra and the intermediate value theorem.
• To formalize Gauss’s geometric notion of winding number in a real-algebraic setting for polynomials.
• To extend Sturm’s real-root counting algorithm to complex roots via Cauchy’s index, using only polynomial arithmetic.
• To develop a fully algorithmic, computer-verifiable root-finding procedure for complex polynomials over any real closed field.
• To recover Brouwer’s fixed point theorem for polynomial maps over real closed fields using the same algebraic winding number framework.

Proposed method

• Formalizing the winding number of a polynomial path around the origin using algebraic topology concepts within real-closed fields.
• Applying Sturm chains to compute the Cauchy index, which counts the number of complex roots within a rectangular domain in the complex plane.
• Using Euclidean division to recursively compute Sturm sequences for both real and complex polynomials, enabling root localization.
• Establishing homotopy invariance of the algebraic winding number to prove topological results like the existence of fixed points.
• Translating the proof into a first-order logical framework valid over all real closed fields, ensuring broad applicability.
• Constructing a root-finding algorithm based on successive subdivision of rectangles, guided by non-zero winding numbers.

Experimental results

Research questions

• RQ1 Can the Fundamental Theorem of Algebra be proven constructively using only real algebra and the intermediate value theorem?
• RQ2 How can Gauss’s geometric winding number be formalized in a purely algebraic and first-order setting?
• RQ3 Can Sturm’s real-root counting algorithm be extended to complex roots using algebraic methods?
• RQ4 What is the computational complexity and effectiveness of the resulting root-finding algorithm over real closed fields?
• RQ5 Can topological results like Brouwer’s fixed point theorem be derived constructively from algebraic winding numbers?

Key findings

• The Fundamental Theorem of Algebra is proven constructively using only the intermediate value theorem and polynomial arithmetic, valid over any real closed field.
• The algebraic winding number provides a real-algebraic equivalent of the topological winding number, enabling rigorous computation of root counts.
• The Sturm–Cauchy algorithm effectively computes the number of complex roots in any rectangle using only polynomial division and sign variations.
• The proof yields a direct, implementable algorithm for locating all complex roots of a polynomial, with convergence guaranteed by successive subdivision.
• The method extends to proving Brouwer’s fixed point theorem for polynomial maps over real closed fields via homotopy invariance of the winding number.
• The entire proof can be formalized in first-order logic, making it amenable to computer verification over all real closed fields.