160 citations · 199 across the 25 of their papers we have counts for
29 papers
Strongly regular graph, strongly polynomial sequences, and two-level polynomials
Tristram Bogart, Santiago Jiménez-Salazar
We study the behavior of the characteristic and chromatic polynomials of certain families of graphs and their relation to homomorphism counts, focusing on polynomial dependence of…
Optimal Small Set Expanders and Their Codes
Tristram Bogart, Marcelo Fiori, Pedro Raigorodsky +1
A left-regular bipartite graph of degree is called a -small-set-expander if every subset of left vertices of size at most has at least neighbors. Such…
A differential characterization of volume polynomials of permutohedra
Tristram Bogart, Federico Castillo, Damián de la Fuente +1
We study a graded vector space of polynomials associated to a square matrix, defined by a finite difference condition along the rows. We show this space coincides with one defined…
Counting with two-level polynomials
Tristram Bogart, Kevin Woods
We examine combinatorial counting functions with two parameters, and . For fixed , these functions are (quasi-)polynomial in . As varies, the degree of this polyno…
Improved Upper Bounds on Key Invariants of Erdős-Rényi Numerical Semigroups
Tristram Bogart, Santiago Morales
De Loera, O'Neill and Wilburne introduced a general model for random numerical semigroups in which each positive integer is chosen independently with some probability p to be a gen…
Unboundedness of irreducible decompositions of numerical semigroups
Tristram Bogart, Seyed Amin Seyed Fakhari
We present two families of numerical semigroups and show that for each family, the number of required components in an irreducible decomposition cannot be bounded by any given inte…
An Ehrhart theoretic approach to generalized Golomb rulers
Tristram Bogart, Daniel Felipe Cuéllar
A Golomb ruler is a sequence of integers whose pairwise differences, or equivalently pairwise sums, are all distinct. This definition has been generalized in various ways to allow…
Numerical semigroups via projections and via quotients
Tristram Bogart, Christopher O'Neill, Kevin Woods
We examine two natural operations to create numerical semigroups. We say that a numerical semigroup is -normalescent if it is the projection of the set of integer…
When is a numerical semigroup a quotient?
Tristram Bogart, Christopher O'Neill, Kevin Woods
A natural operation on numerical semigroups is taking a quotient by a positive integer. If is a quotient of a numerical semigroup with generators, we call $\mathca…
Bounds on Determinantal Complexity of Two Types of Generalized Permanents
Tristram Bogart, Juan Andrés Valero
We define two new families of polynomials that generalize permanents and prove upper and lower bounds on their determinantal complexities comparable to the known bounds for permane…
Complex psd-minimal polytopes in dimensions two and three
Tristram Bogart, João Gouveia, Juan Camilo Torres
The extension complexity of a polytope measures its amenability to succinct representations via lifts. There are several versions of extension complexity, including linear, real se…
A Plethora of Polynomials: A Toolbox for Counting Problems
Tristram Bogart, Kevin Woods
A wide variety of problems in combinatorics and discrete optimization depend on counting the set of integer points in a polytope, or in some more general object constructed via…
Constructing Partial MDS Codes from Reducible Curves
Tristram Bogart, Anna-Lena Horlemann-Trautmann, David Karpuk +2
We propose reducible algebraic curves as a mechanism to construct Partial MDS (PMDS) codes geometrically. We obtain new general existence results, new explicit constructions and im…
An Algebraic Approach to Projective Uniqueness with an Application to Order Polytopes
Tristram Bogart, João Gouveia, Juan Camilo Torres
A combinatorial polytope is said to be projectively unique if it has a single realization up to projective transformations. Projective uniqueness is a geometrically compelling…
Periodic behavior in families of numerical and affine semigroups via parametric Presburger arithmetic
Tristram Bogart, John Goodrick, Kevin Woods
Let be polynomial functions of . For fixed , let be the numerical semigroup generated by $f_1(n),\ldots,f_k(n)…
A parametric version of LLL and some consequences: parametric shortest and closest vector problems
Tristram Bogart, John Goodrick, Kevin Woods
Given a parametric lattice with a basis given by polynomials in Z[t], we give an algorithm to construct an LLL-reduced basis whose elements are eventually quasi-polynomial in t: th…
An switch trick to obtain a new elementary proof of a criterion for arithmetic equivalence
Tristram Bogart, Guillermo Mantilla-Soler
Two number fields are called arithmetically equivalent if they have the same Dedekind zeta function. In the 1970's Perlis showed that this is equivalent to the condition that for a…
Parametric Presburger Arithmetic: Complexity of Counting and Quantifier Elimination
Tristram Bogart, John Goodrick, Danny Nguyen +1
We consider an expansion of Presburger arithmetic which allows multiplication by parameters . A formula in this language defines a parametric set $S_\mathbf{t}…
Parametric Presburger arithmetic: logic, combinatorics, and quasi-polynomial behavior
Tristram Bogart, John Goodrick, Kevin Woods
Parametric Presburger arithmetic concerns families of sets S_t in Z^d, for t in N, that are defined using addition, inequalities, constants in Z, Boolean operations, multiplication…
A lower bound for the determinantal complexity of a hypersurface
Jarod Alper, Tristram Bogart, Mauricio Velasco
We prove that the determinantal complexity of a hypersurface of degree is bounded below by one more than the codimension of the singular locus, provided that this codimensi…
Superlinear subset partition graphs with dimension reduction, strong adjacency, and endpoint count
Tristram C. Bogart, Edward D. Kim
We construct a sequence of subset partition graphs satisfying the dimension reduction, adjacency, strong adjacency, and endpoint count properties whose diameter has a superlinear a…
Hom-polytopes
Tristram Bogart, Mark Contois, Joseph Gubeladze
We study the polytopes of affine maps between two polytopes -- the hom-polytopes. The hom-polytope functor has a left adjoint -- tensor product polytopes. The analogy with the cate…
Enumeration of Golomb Rulers and Acyclic Orientations of Mixed Graphs
Matthias Beck, Tristram Bogart, Tu Pham
A \emph{Golomb ruler} is a sequence of distinct integers (the \emph{markings} of the ruler) whose pairwise differences are distinct. Golomb rulers can be traced back to additive nu…
Obstructions to lifting tropical curves in surfaces in 3-space
Tristram Bogart, Eric Katz
Tropicalization is a procedure that takes subvarieties of an algebraic torus to balanced weighted rational complexes in space. In this paper, we study the tropicalizations of curve…