Problems relating to arcs in projective space and subrings in Z^n

An important problem in mathematics is understanding the behavior of functions. If we can show that complicated functions can be simplified, computation becomes more feasible. In this thesis, we describe two major projects: counting arcs in projective space and counting subrings in $\Z^n$. In the case of arcs in $\mb{P}^{k-1}(\Fq)$, we seek to understand whether the counting function can be expressed by a simple formula like a polynomial. In the case of subrings of $\Z^n$, we are interested in the asymptotic growth of the function that counts subrings of index at most $X$. An $n$-arc in $(k-1)$-dimensional projective space is a set of $n$ points so that no $k$ lie on a hyperplane. Let $C_{n,k}(q)$ be the number of $n$-arcs in $\mb{P}^{k-1}(\Fq)$. We discuss new results for $n$-arcs when $k = 3$ and 4. Building off work of Kaplan, Kimport, Lawrence, Peilen, and Weinreich, we show that $C_{10,3}(q)$ is not quasipolynomial in $q$. For almost all $n$, no formulas for $C_{n,k}(q)$ were known when $k \ge 4$. We introduce a new algorithm to count $n$-arcs in $\mb{P}^3(\Fq)$ in terms of a small number of special combinatorial objects and use it to compute $C_{n,4}(q)$ for $n \le 7$. Finally, we discuss generalizations of this algorithm to higher-dimensional projective space.Let $f_n(p^e)$ count the number of subrings of index $p^e$ in $\Z^n$. We study lower bounds for $f_n(p^e)$ using techniques from combinatorics and arithmetic geometry. Using these lower bounds, we give the best known result about the asymptotic growth of subrings in $\Z^n$ when $n \ge 8$ by studying the analytic properties of the subring zeta function for $\Z^n$. These results immediately give new information about the asymptotic growth of orders in a fixed degree $n$ number field. Finally, we consider the behavior of $f_n(p^e)$ as a function of $p$. We give evidence that this function may always be polynomial in $p$.