An Interesting Family of Symmetric Polynomials

Abstract

We discuss two natural extremal problems for homogeneous polynomials. These problems have simple solutions for polynomials in one or two variables but become interesting for polynomials in three or more variables. We introduce a family of homogeneous symmetric polynomials in three variables that solve one of these problems and have a number of other interesting properties. For example, their coefficients are integers that can be expressed as sums of binomial coefficients and possess a certain divisibility property. Furthermore, these polynomials are connected in a simple way to a family of polynomials arising as sharp examples in the study of proper polynomial mappings between balls in complex Euclidean space.

• MSC: Primary 32H35
• Secondary 11C08

Acknowledgments

The author acknowledges support from NSF grant DMS 1200815. The author would also like to thank John D’Angelo for his encouragement.

Additional information

Notes on contributors

Jennifer Brooks

JENNIFER BROOKS earned her Ph.D. in Mathematics from the University of Wisconsin in 2005. In 2005, she joined the faculty at the University of Montana. Her research interests include harmonic analysis and the theory of functions of several complex variables.