Geometric constructions over finite fields, elementary equivalence of finitely generated fields, and rational points on varieties

有限域上的几何构造、有限生成域的基本等价以及簇上的有理点

基本信息

批准号：
0301280
负责人：
Bjorn Poonen
金额：
$ 37.54万
依托单位：
University of California-Berkeley
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2003
资助国家：
美国
起止时间：
2003-07-01 至 2008-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0301280&HistoricalAwards=false
关键词：
Geometric constructions over finite fields

项目摘要

DMS-0301280Bjorn M. PoonenThe investigator proposes:1) to use sieve methods to prove the existence of various geometricobjects and maps over finite fields. For many problems, the standarddimension-based arguments work only over infinite fields; the idea isto replace them by numerical counting.2) to continue the study initiated by F. Pop on the question ofwhether elementarily equivalent finitely generated fields arenecessarily isomorphic.3) to write a graduate text highlighting the application of schemes tothe study of varieties over nonalgebraically closed fields.Originally, algebraic geometry was concerned with the real numbersolutions to systems of polynomial equations, but the 20th centurymade it clear that it was fruitful to generalize by considering fields(i.e., number systems) other than the real numbers. For example,finite fields have proved invaluable in the theory of error-correctingcodes and cryptography. Unfortunately, many geometric constructionsare known to work only over infinite fields. The investigator's firstproject is to use combinatorial techniques to adapt theseconstructions to the finite field case. The investigator's secondproject concerns the relationships between geometry over differentfields; potentially it could allow results over one field to betransferred to another field. Finally, the investigator proposes towrite a graduate text that will make newly developed techniques inthis area accessible to students.

DMS-0301280BJORN M. POONTIONTHE研究者建议：1）使用筛子方法证明在有限磁场上存在各种几何图和地图。对于许多问题，基于标准的论点仅在无限领域上起作用。 2）为了继续由F. Pop提起的研究，对基本等效产生的领域的问题进行了不必要的同构。第20个百分库很明显，通过考虑实际数字以外的田地（即数字系统）来概括。例如，在误差校正码和加密理论中，有限领域已被证明是无价的。不幸的是，许多已知仅在无限领域工作的几何结构。研究者的第一项项目是使用组合技术将这些构造适应有限的现场情况。研究者的二次项目涉及不同场地上几何之间的关系。它可能会允许一个字段的结果与另一个字段合并。最后，调查员提出了拖曳文本，该文本将使学生可以在这个区域中获得新开发的技术。