A Convex-Analytical Proof of the Fundamental Theorem of Algebra
Functional Analysis
2025-04-10 v1 Spectral Theory
Abstract
A weak version of Birkhoff's generalization of the Perron-Frobenius theorem states that every endomorphism of a finite-dimensional real vector that leaves invariant a non-degenerate closed convex cone has an eigenvector in that cone. Here, we show that this theorem, whose proof relies only upon basic convex analysis, yields very short proofs of both the spectral theorem for selfadjoint operators of Euclidean spaces and the Fundamental Theorem of Algebra.
Keywords
Cite
@article{arxiv.2504.06695,
title = {A Convex-Analytical Proof of the Fundamental Theorem of Algebra},
author = {Clément de Seguins Pazzis},
journal= {arXiv preprint arXiv:2504.06695},
year = {2025}
}
Comments
9 pages (including an appendix with an elementary proof of Birkhoff's theorem)