Complex Axis and de Medeiros' Campo Vetorial

TL;DR

This paper presents a dual geometric proof of the Complex Axis theorem for complex vector spaces, avoiding determinant calculations and leveraging topological and bordism concepts to establish the existence of eigenvectors.

Contribution

It offers a more uniform, topological proof of the Complex Axis theorem that does not rely on Euler characteristics or Lefschetz numbers, using vector fields and bordism techniques.

Findings

Provides a dual geometric proof of the Complex Axis theorem

Shows all vector fields of de Medeiros type are co-bordant to Milnor-Hopf vector fields

Derives the main theorem on complex polynomials using this geometric approach

Abstract

The Complex Axis theorem states that any endomorphism of a finite-dimensional complex vector space affords an eigen-vector (or "invariant axis"). A geometric proof of this geometric result was given by A. de Medeiros, transforming the endomorphism into a topological self-map with Lefschetz number not equal to zero. We give a dual version of this proof, which may be more uniform, and does not rely on the need to do any calculation of an Euler characteristic or Lefschetz number. A vector field on Projective space is read off directly from the coordinates ("entries") of the given endomorphism (complex square matrix). A bordism is defined between such vector fields by means of Stokes' Theorem applied to a real manifold-with-boundary. This is the principle behind Hopf's lemma relating the Gauss map and the index of a vector field. All vector fields of the de Medeiros type are co-bordant to the Milnor-Hopf vector field. This latter comes from a non-derogatory, real diagonal endomorphism, so clearly possesses an eigen-vector. Therefore so has the given arbitrary endomorphism. The main theorem on complex polynomials naturally follows, using the companion matrix, secular polynomial reciprocity. The geometric Complex Axis derivation is meant to avoid determinants or "general position" arguments.