Polynomial Roots and Open Mappings

TL;DR

This paper explores the openness of complex polynomial mappings and applies it to provide new elementary proofs of the Fundamental Theorem of Algebra, emphasizing topological and geometric insights with minimal reliance on derivatives.

Contribution

It offers novel elementary proofs of polynomial openness and the FTA, utilizing topological transversality and analytic mapping techniques, simplifying previous complex arguments.

Findings

New elementary proofs of polynomial openness.

Clarification of algebraic curve behavior in domains.

Finiteness of extrema due to Bézout's theorem.

Abstract

The "openness" of a complex polynomial mapping is discussed and applied to the Fundamental Theorem of Algebra. In this category fall proofs of S. Wolfenstein, R.L. Thompson, J. Milnor, and S. Reich-S. Smale. These proofs take into account the critical points of the polynomial. New elementary proofs of openness due to D. Reem and F.S. Cater make possible a very short proof of FTA without reference to zeros of the derivative. We regard Gauss's Helmstedt Thesis (1799) as an exercise in applied differential topology, and fill out a synopsis published by S. Gersten and J. Stallings. The polynomial function may be perturbed or re-aligned (work of Martin, Savitt and Singer) to eliminate critical values so we work with a configuration of real, plane algebraic curves. The treatment we give to the Implicit Function Theorem uses contractive operators on the Banach algebra of convergent power series according to W. Walter. In addition to Functional Analysis, the key geometric insight is a topological transversality result that leads to a rigorous demonstration that the plane curves intersect as Gauss and A. Ostrowski said they would. Working in the category of analytic mappings makes uniqueness of "implicit" solutions more transparent, and the polynomial (and harmonic) nature of the curves imposes finiteness on the number of extrema, coming from a well-known application of the Be'zout Theorem. We think this provides a clarification of Gauss's statement that an algebraic curve entering a domain, must leave again, that is more concrete than declaring such a curve to be a "one-dimensional manifold".