The Distribution of Values of Mahler

提案摘要 0088915 这项研究的主要目标是开发一种新的分析方法来研究某些类型的高度函数。 首席研究员考虑的主要高度函数是马勒测度。 传统上，具有复系数的一元多项式的马勒测度是出现在闭合单位圆盘外部的多项式根的绝对值的乘积。 然而，在当前上下文中，将马勒测度视为实数或复数欧几里德空间上的距离函数是很方便的。 更准确地说，欧几里得空间中的向量可以用多项式的系数向量来标识，因此向量的马勒测度就是相应多项式的马勒测度。 从这个角度来看，马勒测度是一个连续函数，并且是 1 次齐次的。也就是说，马勒测度是数几何意义上的距离函数。 此外，欧几里德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（或傅里叶）变换对该分布函数进行分析。 尽管马勒测度是一个非常复杂的矢量坐标函数，但分布函数却出奇地简单。 特别是，首席研究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属性。 例如，单位球的体积相对于马勒度量是有理数。 单位球的表面可以通过具有整数系数的多项式映射来参数化。 首席研究员希望通过修改此处描述的方法来攻击 D.H. Lehmer 关于马勒测度小值的众所周知的猜想。 进一步的项目是发现椭圆马勒测度的类似结果。马勒测度是一种用于解析和代数数论的各种研究的技术工具。 它在某些用于分解大型多项式的计算机算法中具有实际重要性。 这是因为多项式的马勒测度给出了有关多项式不可约因子数量的信息，因此立即限制了任何因式分解算法的复杂性。 多项式是一类非常基本且重要的数学对象，它出现在各种各样的应用中。 多项式的马勒测度提供了有关多项式的有用信息，但其精确的有用性取决于特定的应用。 例如，马勒测度可用于确定某些动力系统的熵（复杂性的粗略测度）。 首席研究员的研究动机是希望更好地理解马勒测度并寻求数论和应用数学的新应用。

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.