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Nathan Kaplan

Publications and source records attributed to Nathan Kaplan.

At least 19 recordsLinked to original sources
arXiv2026

Sandpile groups of random bipartite graphs and families of distributions with the same moments

Recently, there has been significant interest in applying the method of moments developed by Wood and others to study distributions of finite abelian groups that arise in number theory and combinatorics. When the moments do not grow too fast, they determine a unique distribution. We construct large families of distributions that have the same moments. These families include several distributions that arise naturally in the study of sandpile groups of families of random graphs. Wood determined the distribution of Sylow $p$-subgroups of sandpile groups of Erd\H{o}s--R\'enyi random graphs. This was extended by M\'esz\'aros to sandpile groups of random $d$-regular graphs, who observed an interesting special case when $d$ is even and $p = 2$. We study Sylow $p$-subgroups of sandpile groups of random bipartite graphs and similarly find a special case for $p =2$. Although this distribution differs from that of M\'esz\'aros, we show that they have the same moments and fit into our broader construction. To compute the moments of the distributions we study, we apply combinatorial tools from the theory of Hall--Littlewood functions.

math.CO↗
arXiv2026

Geometry-Aware MCTS for Extremal Problems in Combinatorial Geometry

We study certain extremal problems in combinatorial geometry that ask about configurations of points in an $n \times n$ grid that satisfy strict, global geometric constraints. Classical exact solvers suffer from combinatorial explosion for these types of problems, and standard reinforcement learning and transformer-based models struggle with the sparse reward "validity cliff" and quadratic token-consumption limits. To overcome these bottlenecks, we propose a Geometry-Aware Monte Carlo Tree Search (MCTS) framework. Our approach strictly enforces geometric constraints through incremental updates to the feasible action space. For constraints about collections of collinear points, like those that occur in the classic No-Three-in-Line problem (Max-N3IL), this mechanism reduces the constraint checking complexity from $O(n^3)$ to $O(n^2)$. To improve search efficiency, we exploit geometric symmetries in two ways: canonical pruning during node expansion to reduce the branching factor, and symmetric batch transitions to accelerate the discovery of promising configurations. We perform extensive experiments and establish new best-known computational results on five out of six of the problems that we considered. Notably, for Max-N3IL we find configurations of size roughly $1.8 n$ for grids of size $82 \le n \le 119$. For the Smallest Complete Set problem, we find configurations of size roughly $0.95 n$, providing new upper bounds within the tested grids. This work establishes Geometry-Aware MCTS as a highly adaptable framework for discovering novel configurations in combinatorial geometry.

cs.AI↗
arXiv2026

A mod $p$ determinant criterion for Cohen--Lenstra convergence of random $p$-adic matrices with prescribed zero patterns

We study the distribution of cokernels of Haar-random matrices over the $p$-adic integers with prescribed zero patterns, motivated by the Cohen--Lenstra heuristics. A central feature of our approach is that the asymptotic cokernel distribution is governed by the reductions modulo $p$ of these matrices, viewed as random matrices over the finite field $\mathbb{F}_p$. For several families of support patterns arising from stair-shaped zero regions, including general stair-shaped patterns, band matrices, and matrices with two symmetric stair-shaped zero regions, we show that convergence of the cokernel distribution to the Cohen--Lenstra distribution is equivalent to an asymptotic nonsingularity condition over $\mathbb{F}_p$. We further propose a conjecture for general support patterns and give examples showing that analogous rank-$r$ criteria fail for $r\ge 1$.

math.NT↗
arXiv2026

The shape of a random numerical semigroup

We study statistical properties of random numerical semigroups of a given genus. We analyze the graph of a typical numerical semigroup, understood as a function from $\mathbb{N}$ to $\mathbb{N}$. If $S$ is a numerical semigroup of genus $g$, this leads us to consider the collection of points $\left(\frac{k-1}{g-1},\frac{a_k(S)}{g} \right)$ where $1 \le k \le g$ and $a_k(S)$ denotes the $k$th smallest nonzero element of $S$. We show that as $g \rightarrow \infty$, this set of points typically becomes closer to a union of two line segments. We prove analogous results for numerical semigroups ordered by Frobenius number.

math.CO↗
arXiv2025

Shortest self-orthogonal embeddings of binary linear codes

There has been recent interest in the study of shortest self-orthogonal embeddings of binary linear codes, since many such codes are optimal self-orthogonal codes. Several authors have studied the length of a shortest self-orthogonal embedding of a given binary code $\mathcal C$, or equivalently, the minimum number of columns that must be added to a generator matrix of $\mathcal C$ to form a generator matrix of a self-orthogonal code. In this paper, we use properties of the hull of a linear code to determine the length of a shortest self-orthogonal embedding of any binary linear code. We focus on the examples of Hamming codes and Reed-Muller codes. We show that a shortest self-orthogonal embedding of a binary Hamming code is self-dual, and propose two algorithms to construct self-dual codes from Hamming codes $\mathcal H_r$. Using these algorithms, we construct a self-dual $[22, 11, 6]$ code, called the shortened Golay code, from the binary $[15, 11, 3]$ Hamming code $\mathcal H_4$, and construct a self-dual $[52, 26, 8]$ code from the binary $[31, 26, 3]$ Hamming code $\mathcal H_5$. We use shortest SO embeddings of linear codes to obtain many inequivalent optimal self-orthogonal codes of dimension $7$ and $8$ for several lengths. Four of the codes of dimension $8$ that we construct are codes with new parameters such as $[91, 8, 42],\, [98, 8, 46],\,[114, 8, 54]$, and $[191, 8, 94]$.

cs.IT↗
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↗
arXiv2025

On the smallest partition associated to a numerical semigroup

The set of hook lengths of an integer partition $\lambda$ is the complement of some numerical semigroup $S$. There has been recent interest in studying the number of partitions with a given set of hook lengths. Very little is known about the distribution of sizes of this finite set of partitions. We focus on the problem of determining the size of the smallest partition with its set of hook lengths equal to $\mathbb{N}\setminus S$.

math.CO↗
arXiv2025

Ordinarization numbers of numerical semigroups

There has been significant recent interest in studying how the number of numerical semigroups of genus $g$ behaves as a function of $g$. Bras-Amor\'os has shown how to organize the collection of numerical semigroups of genus $g$ into a rooted tree called the ordinarization tree. The ordinarization number of a numerical semigroup $S$ is the length of the path from $S$ back to the root of the tree. We study the problem of counting numerical semigroups of genus $g$ with a fixed ordinarization number $r$. We show how this can be interpreted as a counting problem about integer points in a certain rational polyhedral cone and use ideas from Ehrhart theory to study this problem. We give a formula for the number of numerical semigroups of genus $g$ and ordinarization number $2$, building on the corresponding result of Bras-Amor\'os for ordinarization number $1$. We show that the ordinarization number of a numerical semigroup generated by two elements is equal to the number of integer points in a certain right triangle with rational vertices. We consider the analogous problem for supersymmetric numerical semigroups with more generators. We also study ordinarization numbers of numerical semigroups generated by an interval.

math.CO↗
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 $\mathbb{Z}^n$.

math.NT↗
arXiv2024

Hulls of Projective Reed-Muller Codes

Projective Reed-Muller codes are constructed from the family of projective hypersurfaces of a fixed degree over a finite field $\F_q$. We consider the relationship between projective Reed-Muller codes and their duals. We determine when these codes are self-dual, when they are self-orthogonal, and when they are LCD. We then show that when $q$ is sufficiently large, the dimension of the hull of a projective Reed-Muller code is 1 less than the dimension of the code. We determine the dimension of the hull for a wider range of parameters and describe how this leads to a new proof of a recent result of Ruano and San Jos\'e (2024).

cs.IT↗
arXiv2024

Sums of Weighted Lattice Points of Polytopes

We study the problem of counting lattice points of a polytope that are weighted by an Ehrhart quasi-polynomial of a family of parametric polytopes. As applications one can compute integrals and maximum values of such quasi-polynomials, as well as obtain new identities in representation theory. These topics have been of great interest to Mich\`ele Vergne since the late 1980's. Our new contribution is a result that transforms weighted sums into unweighted sums, even when the weights are very general quasipolynomials. In some cases it leads to faster integration over a polytope. We can create new algebraic identities and conjectures in algebraic combinatorics and number theory.

math.CO↗
arXiv2023

A Goldbach theorem for Laurent series semidomains

A semidomain is a subsemiring of an integral domain. One can think of a semidomain as an integral domain in which additive inverses are no longer required. A semidomain $S$ is additively reduced if $0$ is the only invertible element of the monoid $(S,+)$, while $S$ is additively atomic if the monoid $(S,+)$ is atomic (i.e., every non-invertible element $s \in S$ can be written as the sum of finitely many irreducibles of $(S,+)$). In this paper, we describe the additively reduced and additively atomic semidomains $S$ for which every Laurent series $f \in S[[x^{\pm 1} ]]$ that is not a monomial can be written as the sum of at most three multiplicative irreducibles. In particular, we show that, for each $k \in \mathbb{N}$, every polynomial $f \in \mathbb{N}[x_1^{\pm 1}, \ldots, x_k^{\pm 1}]$ that is not a monomial can be written as the sum of two multiplicative irreducibles provided that $f(1, \ldots, 1) > 3$.

math.NT↗
arXiv2023

Cusps in $\mathbb{C}^3$ with prescribed ramification

We study value semigroups associated to germs of maps $\mathbb{C} \rightarrow \mathbb{C}^3$ with fixed ramification profiles in a distinguished point. We then apply our analysis to deduce that Severi varieties of unicuspidal rational fixed-degree curves with value semigroup ${\rm S}$ in $\mathbb{P}^3$ are often reducible when ${\rm S}$ is either 1) the semigroup of a generic cusp whose ramification profile is a supersymmetric triple; or 2) a supersymmetric semigroup with ramification profile given by a supersymmetric triple. In doing so, we uncover new connections with additive combinatorics and number theory.

math.AG↗
arXiv2023

The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions, I

Let $\ell$ be a prime number and let $E$ and $E'$ be $\ell$-isogenous elliptic curves defined over a finite field $k$ of characteristic $p \ne \ell$. Suppose the groups $E(k)$ and $E'(k)$ are isomorphic, but $E(K) \not \simeq E'(K)$, where $K$ is an $\ell$-power extension of $k$. In a previous work we have shown that, under mild rationality hypotheses, the case of interest is when $\ell=2$ and $K$ is the unique quadratic extension of $k$. In this paper we study the likelihood of such an occurrence by fixing a pair of 2-isogenous elliptic curves $E$, $E'$ over ${\mathbf{Q}}$ and asking for the proportion of primes $p$ for which $E(\mathbf{F}_p) \simeq E'(\mathbf{F}_p)$ and $E(\mathbf{F}_{p^2}) \not \simeq E'(\mathbf{F}_{p^2})$.

math.NT↗
arXiv2022

Enumerating numerical sets associated to a numerical semigroup

A numerical set $T$ is a subset of $\mathbb N_0$ that contains $0$ and has finite complement. The atom monoid of $T$ is the set of $x \in \mathbb N_0$ such that $x+T \subseteq T$. Marzuola and Miller introduced the anti-atom problem: how many numerical sets have a given atom monoid? This is equivalent to asking for the number of integer partitions with a given set of hook lengths. We introduce the void poset of a numerical semigroup $S$ and show that numerical sets with atom monoid $S$ are in bijection with certain order ideals of this poset. We use this characterization to answer the anti-atom problem when $S$ has small type.

math.CO↗
arXiv2022

The Expected Embedding Dimension, type and weight of a Numerical Semigroup

We study statistical properties of numerical semigroups of genus $g$ as $g$ goes to infinity. More specifically, we answer a question of Eliahou by showing that as $g$ goes to infinity, the proportion of numerical semigroups of genus $g$ with embedding dimension close to $g/\sqrt{5}$ approaches $1$. We prove similar results for the type and weight of a numerical semigroup of genus $g$.

math.CO↗