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

Michael Eisermann (1)

1 IF - Institut Fourier

Sturm's famous theorem (1829/35) provides an elegant algorithm to count and locate the real roots of any given real polynomial. In his residue calculus of complex functions, Cauchy (1831/37) assimilated this to an algebraic method to count and locate the complex roots of any given complex polynomial. We give a real-algebraic proof of Cauchy's theorem starting from the mere axioms of a real closed field, without appeal to analysis. This allows us to algebraicize Gauss' geometric argument (1799) and thus to derive a real-algebraic proof of the Fundamental Theorem of Algebra, stating that every complex polynomial of degree $n$ has precisely $n$ complex roots. 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. The latter is sufficiently efficient for moderately sized polynomials, but in its present form it still lags behind Schoenhage's nearly optimal numerical algorithm (1982).

Domaines: Géométrie algébrique [math.AG]; Variables complexes [math.CV]; Analyse numérique [math.NA]

Michael Eisermann. The Fundamental Theorem of Algebra made effective: an elementary real-algebraic proof via Sturm chains. 2009. ⟨hal-00363333⟩