# 25 KAM Mathematical Colloquium

## Prof. ANDRZEJ SCHINZEL

### WARSZAVA

## 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.