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If $\Lambda \subseteq \mathbb{Z}^n$ is a sublattice of index $m$, then $\mathbb{Z}^n/\Lambda$ is a finite abelian group of order $m$ and rank at most $n$. Several authors have studied statistical properties of these groups as we range over all sublattices of index at most $X$. In this paper we investigate quotients by sublattices that have additional algebraic structure. While quotients $\mathbb{Z}^n/\Lambda$ follow the Cohen-Lenstra heuristics and are very often cyclic, we show that if $\Lambda$ is actually a subring, then once $n \ge 7$ these quotients are very rarely cyclic. More generally, we show that once $n$ is large enough the quotient typically has very large rank. In order to prove our main theorems, we combine inputs from analytic number theory and combinatorics. We study certain zeta functions associated to $\mathbb{Z}^n$ and also prove several results about matrices in Hermite normal form whose columns span a subring of $\mathbb{Z}^n$., Comment: 29 pages; with an appendix by Gautam Chinta

Star-product graphs are a natural extension of the Cartesian product, but have not been well-studied. We show that many important established and emerging network topologies, including HyperX, SlimFly, BundleFly, PolarStar, mesh, and torus, are in fact star-product graphs. While this connection was known for BundleFly and PolarStar, it was not for the others listed. We extend a method of constructing maximal and near-maximal sets of edge-disjoint spanning trees on Cartesian products to the star product, thus obtain maximal or near-maximal sets of edge-disjoint spanning trees on new networks of importance, where such sets can improve bandwidth of collective operations and therefore accelerate many important workloads in high-performance computing.

We present PolarStar, a novel family of diameter-3 network topologies derived from the star product of low-diameter factor graphs. PolarStar gives the largest known diameter-3 network topologies for almost all radixes, thus providing the best known scalable diameter-$3$ network. Compared to current state-of-the-art diameter-$3$ networks, PolarStar achieves $1.3\times$ geometric mean increase in scale over Bundlefly, $1.9\times$ over Dragonfly, and $6.7\times$ over {3-D} HyperX. PolarStar has many other desirable properties, including a modular layout, large bisection, high resilience to link failures and a large number of feasible configurations for every radix. We give a detailed evaluation with simulations of synthetic and real-world traffic patterns and show that PolarStar exhibits comparable or better performance than current diameter-3 networks., Comment: 13 pages, published in SPAA 2024

In this paper we present PolarFly, a diameter-2 network topology based on the Erdos-Renyi family of polarity graphs from finite geometry. This is a highly scalable low-diameter topology that asymptotically reaches the Moore bound on the number of nodes for a given network degree and diameter PolarFly achieves high Moore bound efficiency even for the moderate radixes commonly seen in current and near-future routers, reaching more than 96% of the theoretical peak. It also offers more feasible router degrees than the state-of-the-art solutions, greatly adding to the selection of scalable diameter-2 networks. PolarFly enjoys many other topological properties highly relevant in practice, such as a modular design and expandability that allow incremental growth in network size without rewiring the whole network. Our evaluation shows that PolarFly outperforms competitive networks in terms of scalability, cost and performance for various traffic patterns., Comment: In Proceedings of International Conference for High Performance Computing, Networking, Storage, and Analysis (SC) 2022

An $n$-cap in $k$-dimensional projective space is a set of $n$ points so that no three lie on a line. In this note, we provide an algorithm to count the number of $n$-caps in $\mathbb{P}^3(\mathbb{F}_q)$, which follows from our recent paper [9]. We then give exact formulas for the number of $n$-caps when $n \le 7$. The formulas are polynomial in $q$ when $n \le 6$ and quasipolynomial in $q$ when $n = 7$., Comment: 8 pages

It is well-known that for each fixed $n$ and $e$, the number of subgroups of index $p^e$ in $\mathbb{Z}^n$ is a polynomial in $p$. Is this true for \emph{subrings} in $\mathbb{Z}^n$ of index $p^e$? Let $f_n(k)$ denote the number of subrings of index $k$ in $\mathbb{Z}^n$. We can define the subring zeta function over $\mathbb{Z}^n$ to be $\zeta_{\mathbb{Z}^n}^R(s) = \sum_{k \ge 1} f_n(k)k^{-s}$. Is this zeta function uniform? These two questions are closely related. In this paper, we describe what is known about these questions, and we make progress toward answering them in a couple ways. First, we describe the connection between counting subrings of index $p^e$ in $\mathbb{Z}^n$ and counting the solutions to a corresponding set of equations modulo various powers of $p$. We then show that the number of solutions to certain subsets of these equations is a polynomial in $p$ for any fixed $n$. On the other hand, we give an example for which the number of solutions to a certain subset of equations is not polynomial. Finally, we give an explicit polynomial formula for the number of `irreducible' subrings of index $p^{n+2}$ in $\mathbb{Z}^n$., Comment: 34 pages including appendix

Fix positive integers $n,r,d$. We show that if $n,r,d$ satisfy a suitable inequality, then any smooth hypersurface $X\subset \mathbb{P}^n$ defined over a finite field of characteristic $p$ sufficiently large contains a rational $r$-plane. Under more restrictive hypotheses on $n,r,d$ we show the same result without the assumption that $X$ is smooth or that $p$ is sufficiently large., Comment: 8 pages

An $n$ arc in $(k-1)$-dimensional projective space is a set of $n$ points so that no $k$ lie on a hyperplane. In 1988, Glynn gave a formula to count $n$-arcs in the projective plane in terms of simpler combinatorial objects called superfigurations. Several authors have used this formula to count $n$-arcs in the projective plane for $n \le 10$. In this paper, we determine a formula to count $n$-arcs in projective 3-space. We then use this formula to give exact expressions for the number of $n$-arcs in $\mathbb{P}^3(\mathbb{F}_q)$ for $n \le 7$, which are polynomial in $q$ for $n \le 6$ and quasipolynomial in $q$ for $n=7$. Lastly, we generalize to higher-dimensional projective space., Comment: Revised based on referee suggestions. Accepted to Finite Fields and Their Applications

Let $f_n(k)$ be the number of subrings of index $k$ in $\mathbb{Z}^n$. We show that results of Brakenhoff imply a lower bound for the asymptotic growth of subrings in $\mathbb{Z}^n$, improving upon lower bounds given by Kaplan, Marcinek, and Takloo-Bighash. Further, we prove two new lower bounds for $f_n(p^e)$ when $e \ge n-1$. Using these bounds, we study the divergence of the subring zeta function of $\mathbb{Z}^n$ and its local factors. Lastly, we apply these results to the problem of counting orders in a number field., Comment: accepted to JNT, minor edits based on the referee's feedback, typos corrected, 22 pages

Idempotent elements are a well-studied part of ring theory, with several identities of the idempotents in $\mathbb{Z}/m\mathbb{Z}$ already known. Although the idempotents are not closed under addition, there are still interesting additive identities that can be derived and used. In this paper, we give several new identities on idempotents in $\mathbb{Z}/ m\mathbb{Z}$. We relate finite sublattices over $\mathbb{Z}/ k\mathbb{Z}$ for all integers $k$ to an infinite lattice that is embedded in the divisibility lattice on $\mathbb{N}$ and to each other as sublattices of this infinite lattice. Using this relation, we generalize several identities on idempotents in $\mathbb{Z}/m\mathbb{Z}$ to those involving idempotents related to these finite sublattices. Finally, as an application of the above idempotent identities, we derive an algorithm for calculating modular exponentiation over $\mathbb{Z}/ m\mathbb{Z}$.

In this paper, we explore the structure of $\mathbb{Z}/ m\mathbb{Z}$ in terms of its orbits under modular exponentiation, illustrating this with a sequential power graph that is naturally derived from the orbits by connecting elements of $\mathbb{Z}/ m\mathbb{Z}$ in the orbit order in which they appear. We find that this graph has a great deal of fascinating algebraic structure. The connected components are composed of orbits that all share at least one element. The vertex sets of the connected components are shown to depend on the factorization of $m$; in fact, the connected components are completely determined by the units of $\mathbb{Z}/ m\mathbb{Z}$, the idempotents of $\mathbb{Z}/ m\mathbb{Z}$ and the square-free divisors of $m$. Both tails and non-tails of the components can be described explicitly and algebraically in terms of these elements of $\mathbb{Z}/ m\mathbb{Z}$. Finally, a lattice of components can be used to show homomorphisms between the non-tails of any two comparable components in the lattice. This extensive structure is used here to prove an algebraic identity on the roots of an idempotent mod $m$, and may be exploited to prove other identities as well.