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

作者建议：1)用筛法证明有限域上各种几何对象和映射的存在性。对于许多问题，标准的基于维度的论证只适用于无穷域；其想法是用数值计数来代替它们。2)继续F.Pop关于初等等价的有限生成的域是否必然同构的问题的研究。3)写一篇研究生课文，强调在非代数闭域上应用方案来研究簇。最初，代数几何关注的是多项式方程组的实数解，但20世纪清楚地表明，通过考虑域(即数系)而不是实数来推广是卓有成效的。例如，有限域在纠错码理论和密码学中被证明是无价的。不幸的是，许多几何构造法只在无穷域上起作用。研究人员的第一个项目是使用组合技术来使这些结构适应有限域的情况。这位研究人员的第二个项目涉及不同领域的几何关系；它可能允许将一个领域的结果转移到另一个领域。最后，研究人员建议撰写一篇研究生课文，使学生能够接触到这一领域的新技术。

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.