代数整数的性质研究和无理测度的计算

The problems on the properties of algebraic integers and on the rational approximations of some irrational numbers have been the focus in Number Theory and attracted much attention. Many problems are hard and have been open for a long time. The importance to study these problems is both in theory and in applications, such as the applications of Mahler Measure in polynomial factorization, transcendence and diophantine approxmation, also the ergodic theory and knot theory. Using computational results to analyze and study some number theory problems becomes more and more important. One of our main research objectives in this research project is to improve the polynomial search abilities with higher degree and then, to obtain better results for some important measures of the algebraic integers, such as the integer transfinite diameter, Mahler measure, the absolute trace, absolute length, etc.. Further study will also include the problems on the properties of algebraic integers such as the integer Chebyshev problem, Lehmer problem, Schur-Siegel-Smyth problem,and its applicaions in Diophantine equations; We will study also the irrationality measures of some important irrational numbers. To provide theoretical and arithmetical supports to the rational representations of the real numbers, research on the rational approximations of some irrational numbers, values of some special functions, and sums of some series will also be part of this project. The research on both theoretical and computational parts in this project has important scientific significance.

代数整数性质的相关问题和无理数的有理逼近问题一直是数论研究中的重点和难点问题，有不少问题长期悬而未决，相关问题的研究具有重要的理论意义和科学价值，如Mahler测度就对多项式分解、超越性理论和丢番图逼近等经典数论问题以及遍历理论和扭结理论等研究有着诸多应用。现有研究成果表明，利用计算结果对研究对象进行理论分析已成为数论研究的重要手段。在本项目中我们将对整超限直径、Mahler测度、绝对迹、绝对长度等代数整数的重要测度进行计算和研究，进而对整切比雪夫问题、Lehmer问题、Schur-Siegel-Smyth迹问题等代数整数性质的重要问题进行讨论，并将绝对长度的研究成果应用到丢番图方程的讨论中；同时对一些重要无理数的无理测度进行计算，讨论这些无理数以及一些特殊函数值和级数值的有理逼近问题，为实数的整数化表示理论和算法研究提供理论依据和算法基础。项目中的理论研究和算法研究都具有重要的科学意义。

在本项目的执行过程中，我们按计划主要对代数整数的测度与性质、定义在[0,1]区间上的整超限直径的上界、Prouhet-Tarry-Escott问题、无理数的有理逼近问题以及一类算术函数在短区间上均值及其应用展开了讨论。通过研究，我们较大程度地改进了一些相关算法，并得到了一些重要的理论结果。相关结论有助于我们更深入研究诸如Lehmer问题、 Schur-Siegel-Smyth迹问题、 Schinzel-Zassenhaus问题以及算术函数的均值估计问题等相关课题，上述问题都是目前悬而未决的问题。