The Distribution of Values of Mahler

这项研究的主要目的是发展一种新的分析方法来研究某些类型的高度函数。主要研究人员考虑的主要高度函数是马勒测量。传统上，复系数一元多项式的马勒测度是该多项式的根在闭单位圆盘外的绝对值的乘积。然而，在目前的上下文中，将马勒度量视为实或复欧几里德空间上的距离函数是方便的。更确切地说，欧几里得空间中的一个向量是用多项式的系数向量来确定的，所以向量的马勒测度就是相应多项式的马勒测度。从这个意义上讲，马勒测度是一个1次齐次连续函数，也就是说，马勒测度是数几何意义上的距离函数。此外，欧氏空间中马勒测度小于实参数的点集的勒贝格测度是该参数的分布函数。然后通过梅林(或傅立叶)变换对该分布函数进行分析。虽然马勒度量是矢量坐标的一个非常复杂的函数，但分布函数却令人惊讶地简单。特别是，主要研究人员观察到与马勒测度相关的自然几何对象的几个意想不到的算术性质。例如，单位球相对于马勒度量的体积是一个有理数。单位球的表面可以用整系数多项式映射来参数化。首席研究者希望通过修改这里所描述的方法来反驳D.H.Lehmer关于马勒测度小值的著名猜想。另一个项目是发现椭圆Mahler测度的类似结果。Mahler测度是一种技术工具，用于分析和代数数论中的各种研究。它在分解大型多项式的某些计算机算法中具有实际意义。这是因为多项式的马勒测度给出了关于多项式的不可约因子的数目的信息，因此立即对任何因式分解算法的复杂性提供了限制。多项式构成了一类非常基本和重要的数学对象，它出现在各种各样的应用中。多项式的马勒测度提供了关于多项式的有用信息，但其确切的有用性取决于特定的应用。例如，马勒度量可以用来确定某些动力系统的熵(复杂性的粗略度量)。首席研究员的研究是为了更好地理解马勒测度，也是为了在数论和应用数学中寻求新的应用。

Abstract for proposal 0088915The 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.