Matricial Proofs of Some Classical Results about Critical Point Location

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.

• MSC: Primary 26C10
• Secondary 30C15
• 15A60

Acknowledgments

The authors thank the anonymous referees and editor-in-chief Susan Colley for their helpful comments.

Additional information

Notes on contributors

Charles R. Johnson

CHARLES R. JOHNSON graduated from Elkhart (IN) High School in 1966, from Northwestern University, with a degree in Mathematics and Economics, in 1969, and then received his Ph.D. from the California Institute of Technology in 1972. After an NRC/NAS postdoc at the National Bureau of Standards, he took a joint position in the Institute for Physical Science and Technology and the Department of Economics at the University of Maryland in 1974, where he was tenured in 1976. After two years as Professor of Mathematical Sciences at Clemson University, he took the Class of 1961 Professorship of Mathematics at William and Mary in 1987. He has now published well over 400 papers and several books, including Matrix Analysis and Topics in Matrix Analysis (with Roger Horn) and Totally Nonnegative Matrices (with Shaun Fallat). His most recent book, Eigenvalues, Multiplicities and Graphs (with Carlos Saiago), just appeared from Cambridge University Press. Much of his work is in matrix theory and combinatorics, but he has papers in journals of physics, economics, psychology, finance, and statistics, etc. He has won awards such as the Washington Academy of Sciences Award for Outstanding Scientific Achievement and the Virginia Outstanding Faculty Member Award and been editor of several journals. He continues to run a long-standing REU program, for which he welcomes applications from any students deeply interested in mathematics. This has resulted in several dozen publications in high-level journals.

Pietro Paparella

PIETRO PAPARELLA received the Ph.D. degree in mathematics from Washington State University in 2013 under the supervision of Michael Tsatsomeros and Judi McDonald. From 2013 to 2015 he held the position of Visiting Assistant Professor in the Department of Mathematics at the College of William and Mary and since 2015 he has held the position of Assistant Professor in the Division of Engineering and Mathematics at the University of Washington Bothell. His research interests are in nonnegative matrix theory, combinatorial matrix theory, discrete geometry, and the geometry of polynomials. This work is his first in The American Mathematical Monthly.