Lehmer's Problem for Polynomials Having Small Mahler Measure on Bernoulli's Lemniscate Ryan Looney, Oklahoma State University Note that this talk is on a Tuesday, not Thursday.
Abstract: The Mahler measure of a complex polynomial is the geometric mean of that polynomial over the unit circle. By a result of Kronecker, for nonconstant integer polynomials, the Mahler measure is equal to 1 if and only if all roots of the polynomial are 0 or roots of unity. In 1933, Lehmer asked if the Mahler measure for all other nonconstant integer polynomials had a lower bound greater than 1. In fact, Lehmer noted that the smallest such measure he had found belonged to a polynomial having degree 10, and to this day no polynomial has been found which lowers this bound. We explore a generalization of the Mahler measure and Lehmer's problem over the classical Bernoulli's lemniscate and its variations.
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