After studying elementary calculus, one could easily come to the conclusion that there is not much of mathematical interest to be said about polynomials. They are infinitely differentiable everywhere and both differentiation and integration are done by elementary formulas. What else is there to know?

It turns out that there are thousands of research papers on polynomials with beautiful and unexpected results and many new ones appear each year. This week of lectures discusses some of the time honored theorems about polynomials due to Chebyshev, Markov, Bernstein, Rogosinski and Grace. The focus will be on how information on the values of a polynomial determines estimates on the size of the derivative of the polynomial. The main tool will be the Lagrange interpolation formula, which expresses a polynomial of degree n in terms of its values at n+1 points. Some new research results obtained in this way will be presented.

We then turn to a more abstract approach which depends on viewing a polynomial as a function of a complex variable and using results about complex numbers. Here the approach is to apply a theorem of Hormander which shows a connection between the roots of a polynomial and the roots of its derivative. The arguments apply not only in the case of the complex plane but to any complex vector space such as the set of all n x n matrices with complex entries.