The primary objective of this research is to develop a new analytic method for investigating certain types of height functions. The main height function considered by the principal investigator is the Mahler measure. Traditionally, the Mahler measure of a monic polynomial with complex coefficients is the product of the absolute values of those roots of the polynomial which occur outside the closed unit disk. In the present context, however, it is convenient to regard the Mahler measure as a distance function on a real or complex Euclidean space. More precisely, a vector in Euclidean space is identified with the vector of coefficients of a polynomial and so the Mahler measure of the vector is simply the Mahler measure of the corresponding polynomial. Viewed in this way the Mahler measure is a continuous function and homogeneous of degree 1. That is, the Mahler measure is a distance function in the sense of the geometry of numbers. Also, the Lebesgue measure of the set of points in Euclidean space where the Mahler measure is less than a real parameter, is a distribution function of the parameter. This distribution function is then subject to analysis by means of the Mellin (or Fourier) transform. Although the Mahler measure is a very complicated function of the coordinates of a vector, the distribution function is shown to be surprisingly simple. In particular, the principal investigator observes several unexpected arithmetical properties of natural geometric objects associated to the Mahler measure. For example, the volume of the unit ball with respect to the Mahler measure is a rational number. And the surface of the unit ball can be parameterized by polynomial maps with integer coefficients. The principal investigator hopes to attack the well known conjecture of D.H. Lehmer concerning small values of Mahler's measure by a modification of the methods described here. A further project is to discover analogous results for the elliptic Mahler measure.
The Mahler measure is a technical tool used in a variety of investigations in analytic and algebraic number theory. And it has practical importance in certain computer algorithms for factoring large polynomials. This is because the Mahler measure of a polynomial gives information about the number of irreducible factors of the polynomial, and so immediately provides a limit to the complexity of any factoring algorithm. Polynomials form a very basic and important class of mathematical objects which appear in a wide variety of applications. The Mahler measure of a polynomial gives useful information about the polynomial, but its precise usefulness depends on the particular application. For example, the Mahler measure can be used to determine the entropy (a rough measure of complicatedness) of certain dynamical systems. The research of the principal investigator is motivated by a desire to better understand the Mahler measure and also to seek new applications in number theory and in applied mathematics.
