Number Theory in Function Fields

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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.