31 citations · 208 across the 28 of their papers we have counts for
28 papers
Sums of Weighted Lattice Points of Polytopes
Jesús A. De Loera, Laura Escobar, Nathan Kaplan +1
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…
Cusps in with prescribed ramification
Ethan Cotterill, Nathan Kaplan, Renata Vieira Costa
We study value semigroups associated to germs of maps with fixed ramification profiles in a distinguished point. We then apply our analysis to…
The probability of non-isomorphic group structures of isogenous elliptic curves in finite field extensions, I
John Cullinan, Nathan Kaplan
Let be a prime number and let and be -isogenous elliptic curves defined over a finite field of characteristic . Suppose the groups and…
The Expected Embedding Dimension, type and weight of a Numerical Semigroup
Nathan Kaplan, Deepesh Singhal
We study statistical properties of numerical semigroups of genus as goes to infinity. More specifically, we answer a question of Eliahou by showing that as goes to infi…
Enumerating numerical sets associated to a numerical semigroup
April Chen, Nathan Kaplan, Liam Lawson +2
A numerical set is a subset of that contains and has finite complement. The atom monoid of is the set of such that .…
Generalizations of results of Friedman and Washington on cokernels of random -adic matrices
Gilyoung Cheong, Nathan Kaplan
Let be prime and be a Haar-random matrix over , the ring of -adic integers. Let be monic p…
Counting Plane Cubic Curves over Finite Fields with a Prescribed Number of Rational Intersection Points
Nathan Kaplan, Vlad Matei
For each integer , we count the number of plane cubic curves defined over a finite field that do not share a common component and intersect in exactly $…
Numerical semigroups, polyhedra, and posets I: the group cone
Nathan Kaplan, Christopher O'Neill
Several recent papers have explored families of rational polyhedra whose integer points are in bijection with certain families of numerical semigroups. One such family, first intro…
Chip-Firing Games and Critical Groups
Darren Glass, Nathan Kaplan
In this expository article intended to be accessible to undergraduate students we introduce a finite abelian group that can be associated to any finite connected graph. This group…
Weight enumerators of Reed-Muller codes from cubic curves and their duals
Nathan Kaplan
Let be a finite field of characteristic not equal to or . We compute the weight enumerators of some projective and affine Reed-Muller codes of order over…
Random Partitions and Cohen-Lenstra Heuristics
Jason Fulman, Nathan Kaplan
We investigate combinatorial properties of a family of probability distributions on finite abelian p-groups. This family includes several well-known distributions as specialization…
The cotype zeta function of
Gautam Chinta, Nathan Kaplan, Shaked Koplewitz
We give an asymptotic formula for the number of sublattices of index at most for which has rank at most , answering a question of…
Counting Numerical Semigroups
Nathan Kaplan
A numerical semigroup is an additive submonoid of the natural numbers with finite complement. The size of the complement is called the genus of the semigroup. How many numerical se…
Representations of the Multicast Network Problem
Sarah E. Anderson, Wael Halbawi, Nathan Kaplan +4
We approach the problem of linear network coding for multicast networks from different perspectives. We introduce the notion of the coding points of a network, which are edges of t…
Counting Arithmetical Structures on Paths and Cycles
Benjamin Braun, Hugo Corrales, Scott Corry +6
Let be a finite, simple, connected graph. An arithmetical structure on is a pair of positive integer vectors such that $(\mathrm{diag}(\mathbf{d})-A…
Counting Arcs in Projective Planes via Glynn's Algorithm
Nathan Kaplan, Susie Kimport, Rachel Lawrence +2
An -arc in a projective plane is a collection of distinct points in the plane, no three of which lie on a line. Formulas counting the number of -arcs in any finite projec…
Counting Finite Index Subrings of
Stanislav Atanasov, Nathan Kaplan, Benjamin Krakoff +1
We count subrings of small index of , where the addition and multiplication are defined componentwise. Let denote the number of subrings of index . For an…
Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks
Jennifer S. Balakrishnan, Wei Ho, Nathan Kaplan +3
Most systematic tables of data associated to ranks of elliptic curves order the curves by conductor. Recent developments, led by work of Bhargava-Shankar studying the average sizes…
Elliptic curves over a finite field and the trace formula
Nathan Kaplan, Ian Petrow
We prove formulas for power moments for point counts of elliptic curves over a finite field such that the groups of -points of the curves contain a chosen subgroup. These fo…
Numerical Sets, Core Partitions, and Integer Points in Polytopes
Hannah Constantin, Benjamin Houston-Edwards, Nathan Kaplan
We study a correspondence between numerical sets and integer partitions that leads to a bijection between simultaneous core partitions and the integer points of a certain polytope.…
Traces of Hecke Operators and Refined Weight Enumerators of Reed-Solomon Codes
Nathan Kaplan, Ian Petrow
We study the quadratic residue weight enumerators of the dual projective Reed-Solomon codes of dimensions and over the finite field . Our main results are f…
The distribution of -points on cyclic -covers of genus
Alina Bucur, Chantal David, Brooke Feigon +4
We study fluctuations in the number of points of -cyclic covers of the projective line over the finite field when is fixed and the genus…
The Realization Problem for Delta Sets of Numerical Semigroups
Stefan Colton, Nathan Kaplan
The delta set of a numerical semigroup , denoted , is a factorization invariant that measures the complexity of the sets of lengths of elements in . We study the follow…
Distribution of orders in number fields
Nathan Kaplan, Jake Marcinek, Ramin Takloo-Bighash
In this paper we study the distribution of orders of bounded discriminants in number fields. We give an asymptotic formula for the number of orders contained in the ring of integer…