The Distribution of Values of Mahler

本研究的主要目的是开发一种新的分析方法来研究某些类型的高度函数。主要研究者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在封闭单位圆盘外出现的多项式的根的绝对值的乘积。然而，在目前的情况下，将马勒测度视为实或复欧几里得空间上的距离函数是方便的。更准确地说，欧几里得空间中的一个向量可以用多项式的系数向量来标识，所以这个向量的马勒测度就是对应多项式的马勒测度。这样看来，马勒测度是一个1次齐次的连续函数。也就是说，马勒测度是数字几何意义上的距离函数。同样，欧几里得空间中马勒测度小于实参数的点集的勒贝格测度，是参数的分布函数。这个分布函数然后通过梅林（或傅里叶）变换进行分析。虽然马勒测度是矢量坐标的一个非常复杂的函数，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物体的几个意想不到的算术性质。例如，单位球的体积相对于马勒测度是一个有理数。单位球的表面可以用整数系数的多项式映射参数化。首席研究员希望通过对这里描述的方法的修改来攻击D.H. Lehmer关于马勒测度的小值的著名猜想。进一步的项目是发现椭圆马勒测度的类似结果。马勒测度是在分析数论和代数数论的各种研究中使用的技术工具。它在某些大型多项式分解的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多项式不可约因子的数量信息，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和重要的数学对象，有着广泛的应用。多项式的马勒测度给出了关于多项式的有用信息，但是它的精确用途取决于特定的应用。例如，马勒测量可以用来确定某些动力系统的熵（复杂程度的粗略测量）。首席研究员的研究动机是为了更好地理解马勒测度，并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.