Diophantine Problems in Function Fields

Abstract: Let $\mathbb{Z}$ be the ring of integers, and let $\mathbb{F}_p[t]$ be the ring of polynomials in one variable defined over the finite field $\mathbb{F}_p$ of $p$ elements. Since the characteristic of $\mathbb{Z}$ is $0$, while that of $\mathbb{F}_p[t]$ is the positive prime number $p$, it is an interesting phenomenon in arithmetic that these two rings resemble one another so faithfully. The study of the similarity and difference between $\mathbb{Z}$ and $\mathbb{F}_p[t]$ lies in the field that relates number fields to function fields. In this talk, we will investigate some Diophantine problems in the settings of $\mathbb{Z}$ and $\mathbb{F}_p[t]$, including Waring's problem about representations of elements with fixed powers.

number theory

Audience: researchers in the discipline

K-State Mathematics Department Women Lecture Series