Number Theory in Function Fields

数论是关于整数环Z的算术性质的研究。我们可以将这项研究推广到其他具有环状结构的物体上。设Fq[t]是一个一元多项式环，定义在有q个元素的有限域Fq上。Z和Fq[t]之间惊人的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]的特征是一个正素数。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场中的数论。更具体地说，我应用圆方法来研究Fq[t]-可加数论问题的类似物，如Waring问题和Vinogradov的中值定理。研究了Weyl在函数场中的等分布定理，并将其应用于加性组合问题的研究。此外，我还使用筛法研究解析数论中的问题，如Drinfeld模块的Erdos-Pomerance猜想。****** **** **

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. ****** **** **