Locating Unimodular Roots

Year of Award: 2015

Award: Pólya

Publication Information: College Mathematics Journal, vol. 45, no. 3, May 2014, pp. 162-168.

Summary (adapted from the MAA Prizes and Awards booklet for MathFest 2015): How do the roots of the polynomial p(z) = zⁿ − 1 change when a lower order term z^k is added to it? Given values of n and k, with 1 ≤ k ≤ n, how many roots of the trinomial q(z) = zⁿ + z^k − 1, if any, still lie on the unit circle? In this intriguing article, Brilleslyper and Schaubroeck prove that q has unimodular roots (roots that lie on the unit circle) if and only if 6 divides (n + k)/gcd(n, k) and that when this divisibility condition is satisfied, q has 2gcd(n, k) unimodular roots. In addition, they determine the exact locations of these roots. Their main result relies on a classical theorem about Diophantine equations, thus forming a nice connection between number theory and complex analysis.

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About the Authors: (From the MathFest 2015 MAA Prizes and Awards Booklet)

Michael Brilleslyper earned his Ph.D. at the University of Arizona in 1994. He spent the first six years of his career at Arizona State University, where he was heavily involved in the First-Year Mathematics Program. The years at ASU were instrumental in developing his appreciation for the art of teaching and the importance of training and mentoring new instructors. Michael has been at the Air Force Academy since 2000. He has participated extensively in pedagogy and curriculum design as well as faculty development, and has served as academic strategic advisor. He enjoys working on a variety of mathematical projects to include polynomial roots, integer sequences, and other topics that include the potential for undergraduate research. Mike is celebrating 20 years of MAA membership and the Association has played an important role in his career. He was a Project NExT fellow in 1995, chairman of the Rocky Mountain Section, Governor of the Section, has served on a number of other committees and projects. Mike and his wife MaryAnn have been married for 26 years, and have two wonderful daughters who keep their lives exciting and fun.

Lisbeth Schaubroeck earned her Ph.D. from the University of North Carolina at Chapel Hill in 1998 under the direction of the late John Pfaltzgraff, who she knows would have been very proud to have his student win this award. He would have humbly said that he had nothing to do with it, but Beth knows better—John was the first person to help her write careful mathematics. Beth’s entire academic career has been at the United States Air Force Academy in Colorado Springs. In addition to dabbling in the study of polynomial roots, she has also recently published articles related to surfaces associated with iterated functions, undergraduate teaching, and the mathematics of the wind. At the Air Force Academy, she has worked as faculty development director for her department, co-coordinator for the mathematics major, and advisor for academic strategy. She enjoys teaching courses across the curriculum, from freshman calculus to senior complex analysis, and has mentored many student projects in elementary knot theory. This year she and her husband Tim celebrated their 25th wedding anniversary; she is grateful for Tim’s continued support of her career. They have two sons, who keep their parents busy with robotics, archery, baseball, basketball, and music. Beth seems not to have picked up their athletic skill, but she does enjoy archery and bowling.