Matricial Proofs of Some Classical Results about Critical Point Location. Issue 1 (2nd January 2020)

• Record Type: Journal Article
• Authors: Johnson, Charles R.; Paparella, Pietro
• Is Part Of: American Mathematical Monthly. Volume 127:Issue 1(2020)
• Page Start: 45
• Page End: 53
• Publication Date: 2020-01-02
• Subjects: MSC: Primary 26C10 -- Secondary 30C15 -- 15A60; Mathematics -- Periodicals; 510.5
• DOI: 10.1080/00029890.2020.1671740
• Languages: English
• ISSNs: 0002-9890

Abstract

Abstract: The Gauss–Lucas and Bôcher–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.