25 KAM Mathematical Colloquium

THE MAHLER MEASURE OF POLYNOMIALS IN SEVERAL VARIABLES

March 15, 1996
Lecture Room S6, Charles University, Malostranske nam. 25, Praha 1
10:30 AM

Abstract

For a polynomial $F\in {\bf C}[z_1,\ldots,z_n]-\{0\}$ the Mahler measure $M(F)$ is defined by the formula $$M(F)=\exp \int_{0}^{1}\ldots \int_{0}^{1}\log |F(e(\theta_1),\ldots, e(\theta_s))|d\theta_1\ldots d\theta_s,$$ where $e(\theta)=\exp 2\pi i\theta$. It follows that $$M(F_1F_2)=M(F_1)M(F_2)$$ and if $s=1$, $F=a\prod_{j=1}^k(z-\alpha_j)$ then $$M(F)=|a|\prod_{j=1}^k\max \{1,|\alpha_j|\}.$$ The relation of $M(F)$ to other measures of $F$ will be discussed and some open problems proposed concerning $M(F)$ for $F$ with complex or integral coefficients.

This colloquium is organized by Department of Applied Mathematics (KAM) of Charles University jointly with University of Ostrava.