# Mathematical Sciences Research Institute

Home » Higher Mahler measure and Lehmer's question

# Seminar

Higher Mahler measure and Lehmer's question April 25, 2011 (01:00 PM PDT - 02:00 PM PDT)
Parent Program: -- MSRI: Simons Auditorium
Speaker(s) Kaneenika Sinha (Indian Institute of Science Education and Research)
Description No Description
Video
The Mahler measure M(f) of a monic polynomial f is defined to be the absolute value of the product of those roots of f which lie outside the unit disk. The logarithmic Mahler measure m(f) = log M(f) turns out to be the integral of log|f| on the unit circle. In 1933, Lehmer essentially asked the following question: for any C >0, can we find a polynomial with integer coefficients such that 0< C? Lehmer's question is still open. He constructed a polynomial of degree 10 with m(f) = 0.16235, which still remains the polynomial with the lowest known Mahler measure. In this talk, we will consider higher analogues of the Mahler measure. More precisely, for $k>1,$ we define the k- higher Mahler measure to be the integral of log^k |f| on the unit circle. We explore the analogues of Lehmer's question for these higher Mahler measures. This is joint work with Matilde Lalin.