arXiv subjects

Kelly Isham

Publications and source records attributed to Kelly Isham.

14 recordsLinked to original sources

arXiv2026

Kronecker Products, Polarity Quotients and Large Graph Constructions

In this paper, we establish a structural compatibility between the Kronecker product of bipartite graphs that admit polarity and their polarity quotient, and provide a sharp upper bound on the diameter of these graphs. For certain factor graphs, the diameter of the Kronecker product meets the upper bound on diameter, among them the generalized polygons. Generalized polygons with their polarity quotients have been notably used in the past to construct very large graphs. We apply the structural theorems in the paper to generalized polygons $\mathbb{G}_n(q,q)$ used as factor graphs, and build three new families of graphs of large order covering an infinite but sparse set of degrees, one of diameter $2$, one of diameter $3$ and one of diameter $5$. These asymptotically approach a theoretical upper bound on graph size as orders $q$ and $r$ of the generalized polygon factors increase. As an example, we develop one such family, derived from generalized quadrangles, and construct new diameter-$3$ graphs of low degree that are larger than any previously known at their degrees.

math.CO↗

arXiv2025

Configurations of 10 points and their incidence varieties

Incidence varieties are spaces of $n$-tuples of points in the projective plane that satisfy a given set of collinearity conditions. We classify the components of incidence varieties and realization moduli spaces associated to configurations of up to 10 points, up to birational equivalence. We show that each realization space component is birational to a projective space, a genus 1 curve, or a K3 surface. To do this, we reduce the problem to a study of 163 special arrangements called superfigurations. Then we use computer algebra to describe the realization space of each superfiguration.

math.AG↗

arXiv2024

Most subrings of $\mathbb{Z}^n$ have large corank

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 $\mathbb{Z}^n$.

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arXiv2024

Edge-Disjoint Spanning Trees on Star-Product Networks

A star-product operation may be used to create large graphs from smaller factor graphs. Network topologies based on star-products demonstrate several advantages including low-diameter, high scalability, modularity and others. Many state-of-the-art diameter-2 and -3 topologies~(Slim Fly, Bundlefly, PolarStar etc.) can be represented as star products. In this paper, we explore constructions of edge-disjoint spanning trees~(EDSTs) in star-product topologies. EDSTs expose multiple parallel disjoint pathways in the network and can be leveraged to accelerate collective communication, enhance fault tolerance and network recovery, and manage congestion. Our EDSTs have provably maximum or near-maximum cardinality which amplifies their benefits. We further analyze their depths and show that for one of our constructions, all trees have order of the depth of the EDSTs of the factor graphs, and for all other constructions, a large subset of the trees have that depth.

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arXiv2024

Generalized Eckardt points on del Pezzo surfaces of degree 1

We study intersections of exceptional curves on del Pezzo surfaces of degree 1, motivated by questions in arithmetic geometry. Outside characteristics 2 and 3, at most 10 exceptional curves can intersect in a point. We classify the different ways in which 10 exceptional curves can intersect, construct a new family of surfaces with 10 exceptional curves intersecting in a point, and discuss strategies for finding more such examples.

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arXiv2023

PolarFly: A Cost-Effective and Flexible Low-Diameter Topology

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.

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arXiv2023

PolarStar: Expanding the Scalability Horizon of Diameter-3 Networks

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.

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arXiv2022

An algorithm to count the number of caps in $\mathbb{P}^3(\mathbb{F}_q)$

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 $n = 7$.

math.CO↗

arXiv2022

Is the number of subrings of index $p^e$ in $\mathbb{Z}^n$ polynomial in $p$?

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 $ζ_{\mathbb{Z}^n}^R(s) = \sum_{k \ge 1} f_n(k)k^{-s}$. 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 $\mathbb{Z}^n$.

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arXiv2022

An algorithm for counting arcs in higher-dimensional projective space

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.

math.CO↗

arXiv2021

Rational linear subspaces of hypersurfaces over finite fields

Let $X \subset \mathbb{P}^n$ be a hypersurface of degree $d$ defined over a finite field of characteristic $p > 0$. We prove that if $n \ge r + \binom{d+r}{r+1}$, then $X$ contains a rational $r$-plane. We prove better bounds when $X$ is smooth and $p$ is sufficiently large. We also present experimental data regarding the existence of rational lines on cubic threefolds over $\mathbb{F}_7$, $\mathbb{F}_8$, and $\mathbb{F}_9$. In particular, we construct an example of a smooth cubic threefold over $\mathbb{F}_7$ with exactly $8$ rational lines. It remains an open question whether smooth cubic threefolds over $\mathbb{F}_7$, $\mathbb{F}_8$, and $\mathbb{F}_9$ always contain a rational line.

math.AG↗

arXiv2021

Lower bounds for the number of subrings in $\mathbb{Z}^n$

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.

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arXiv2020

Arithmetic of idempotents in $\mathbb{Z}/m \mathbb{Z}$

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 $\mathbb{Z}/ m\mathbb{Z}$.

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arXiv2020

On structures induced by the power sequences of $($\mathbb{Z}/ m\mathbb{Z}$, \cdot)$

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.

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