Number Theory in Function Fields

函数域中的数论

• 批准号: RGPIN-2016-03720
• 负责人: Liu, YuRu
• 金额: $ 2.4万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=684898
• 关键词: Number; Theory; Function; Fields

Number theory is about the study of arithmetic properties of Z, the ring of integers. One can generalize this study to other objects with the ring structure. Let Fq[t] be the ring of polynomials in one variable defined over the finite field Fq of q elements. The remarkable similarities between Z and Fq[t] is a striking feature of arithmetic. In at least one respect, it is surprising that these rings resemble one another faithfully for the characteristic of Z is 0, while that of Fq[t] is a positive prime number. The analogy between Z and Fq[t] is but one in a family that relates number fields to function fields. My research is on number theory in function fields. More specifically, I apply the circle method to study Fq[t]-analogues of problems in additive number theory, such as Waring's problem and Vinogradov's mean value theorem. I also investigate Weyl's equidistribution theorem in function fields and apply it to study problems in additive combinatorics. In addition, I use sieve methods to study problems in analytic number theory, such as the Erdos-Pomerance conjecture for Drinfeld modules. ****** **** **

数论是关于整数环Z的算术性质的研究。可以将该研究推广到具有环结构的其他对象。设Fq[t]是定义在q元有限域Fq上的一元多项式环。Z和Fq[t]之间的显著相似性是算术的一个显著特征。至少在一个方面，令人惊讶的是，这些环彼此忠实地相似，因为Z的特征是0，而Fq[t]的特征是正素数。Z和Fq[t]之间的类比只是将数域与函数域联系起来的一族中的一个。我的研究方向是函数领域的数论。更具体地说，我应用圆的方法来研究Fq[t]-类似的问题在加法数论，如华林的问题和维诺格拉多夫的中值定理。我还调查外尔的等分布定理在功能领域和应用它来研究问题的添加剂组合。此外，我使用筛子方法来研究解析数论中的问题，例如Drinfeld模的Erdos-Pomerance猜想。****** **** **