Polynomials and Inequalities

MATH 6010

Time and Place: MWF 12:30-1:20 p.m. in MSCS 509
Professor: Igor E. Pritsker
Office: MSCS 519C
Office Hours: MWF 10:30-11:30 a.m.
Office Phone: 744-8220
E-mail: igor@math.okstate.edu
Web: http://www.math.okstate.edu/~igor/
Textbook: V. V. Prasolov, Polynomials, Springer, 2004.

Polynomials are of importance in almost every area of mathematics, so that you are very likely to find this course useful and relevant to your interests. We shall highlight a number of open problems and attractive directions of research.

Most of the of material is rather elementary, and only requires general knowledge of analysis at the level of Advanced Calculus and Elementary Complex Variables. Real Analysis and Complex Analysis courses will definitely suffice as prerequisites.

Each student will work on a project related to the course, and prepare a presentation.

Brief contents

1. General properties of polynomials, connections between their zeros and coefficients.
2. Classical polynomial inequalities, e.g., inequalities of Markov and Bernstein for the derivative.
3. Extremal problems for polynomials, e.g., problems on minimizing various norms that lead to Chebyshev polynomials and orthogonal polynomials.
4. Location of the critical points of polynomials.
5. Applications in Analysis, Number Theory and beyond.

Additional references:
1. Q. I. Rahman and G. Schmeisser, Analytic Theory of Polynomials: Critical Points, Zeros and Extremal Properties, Oxford University Press, 2002.
2. G. V. Milovanovic, D. S. Mitrinovic, Th. M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific, 1994.
3. P. B. Borwein and T. Erdelyi, Polynomials and Polynomial Inequalities, Springer, 1995.
4. P. B. Borwein, Computational Excursions in Analysis and Number Theory, CMS, 2002.
5. G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, Cambridge University Press, 1988.
6. D. S. Mitrinovic, P. M. Vasic, Analytic Inequalities, Springer, 1970.