Matricial Proofs of Some Classical Results about Critical Point Location
Algebraic Geometry
2020-12-24 v1
Abstract
The Gauss--Lucas and B\^{o}cher--Grace--Marden theorems are classical results in the geometry of polynomials. Proofs of the these results are available in the literature, but the approaches are seemingly different. In this work, we show that these theorems can be proven in a unified theoretical framework utilizing matrix analysis (in particular, using the field of values and the differentiator of a matrix). In addition, we provide a useful variant of a well-known result due to Siebeck.
Keywords
Cite
@article{arxiv.2012.12708,
title = {Matricial Proofs of Some Classical Results about Critical Point Location},
author = {Charles R. Johnson and Pietro Paparella},
journal= {arXiv preprint arXiv:2012.12708},
year = {2020}
}
Comments
This is an Accepted Manuscript of an article published by Taylor & Francis in The American Mathematical Monthly on 19-Dec-2019, available at https://www.tandfonline.com/doi/10.1080/00029890.2020.1671740