arXiv subjects

Tristram Bogart

Publications and source records attributed to Tristram Bogart.

At least 19 recordsLinked to original sources

arXiv2026

Counting with two-level polynomials

We examine combinatorial counting functions with two parameters, $n$ and $q$. For fixed $q$, these functions are (quasi-)polynomial in $n$. As $q$ varies, the degree of this polynomial is itself polynomial in $q$, as are the leading coefficients. We carefully define these two-level polynomials, lay out their basic algebraic properties, and provide a schema for showing a function is a two-level polynomial. Using the schema, we prove that a variety of counting functions arising in different areas of combinatorics are two-level polynomials. These include chromatic polynomials for many infinite families of graphs, partitions of an integer into a given number of parts, placing non-attacking chess pieces on a board, Sidon sets, and Sheffer sequences (including binomial type and Appell sequences).

math.CO↗

arXiv2026

Optimal Small Set Expanders and Their Codes

A left-regular bipartite graph $G$ of degree $d$ is called a $(t,α)$-small-set-expander if every subset $X$ of left vertices of size at most $t$ has at least $α|X|$ neighbors. Such a graph is an optimal small-set expander if small subsets have as many neighbors as possible. We characterize optimal expanders combinatorially via girth and prove the existence of $s$-optimal expanders for every $s$. We also prove that $s$-optimality yields new "transfer" lower bounds on the number of neighbors of sets of size $h\geq s$. Finally, as an application, we discuss the use of optimal small-set expanders in building good codes for key exchange protocols in post-quantum cryptography.

math.CO↗

arXiv2026

A differential characterization of volume polynomials of permutohedra

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 by directional derivatives, and prove it is finite-dimensional precisely when all principal minors are nonzero. In that case, its dimension in each degree equals a binomial coefficient, giving total dimension a power of two. For Cartan matrices of irreducible root systems, we construct an explicit basis of volume polynomials of faces of the associated permutohedra, yielding an elementary criterion, which we call geometricity, for expressing a polynomial as a linear combination of these volume polynomials.

math.CO↗

arXiv2025

Improved Upper Bounds on Key Invariants of Erdős-Rényi Numerical Semigroups

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 generator, and proved upper and lower bounds on the expected Frobenius number and expected embedding dimensions. We use a range of probabilistic methods to improve the upper bounds to within a polylogarithmic factor of the lower bounds in each case. As one of the tools to do this, we prove that for any prime q, if A is a random subset of the cyclic group Z_q whose size is of order log(q) and k is also of order log(q), then with high probability the k-fold sumset kA is all of Z_q.

math.AC↗

arXiv2024

Unboundedness of irreducible decompositions of numerical semigroups

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 integer. This gives a negative answer to a question raised by Delgado, García-Sánchez and Rosales.

math.AC↗

arXiv2024

Numerical semigroups via projections and via quotients

We examine two natural operations to create numerical semigroups. We say that a numerical semigroup $\mathcal{S}$ is $k$-normalescent if it is the projection of the set of integer points in a $k$-dimensional polyhedral cone, and we say that $\mathcal{S}$ is a $k$-quotient if it is the quotient of a numerical semigroup with $k$ generators. We prove that all $k$-quotients are $k$-normalescent, and although the converse is false in general, we prove that the projection of the set of integer points in a cone with $k$ extreme rays (possibly lying in a dimension smaller than $k$) is a $k$-quotient. The discrete geometric perspective of studying cones is useful for studying $k$-quotients: in particular, we use it to prove that the sum of a $k_1$-quotient and a $k_2$-quotient is a $(k_1+k_2)$-quotient. In addition, we prove several results about when a numerical semigroup is not $k$-normalescent.

math.AC↗

arXiv2023

An Ehrhart theoretic approach to generalized Golomb rulers

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 for sums of h integers, or to allow up to g repetitions of a given sum or difference. Beck, Bogart, and Pham applied the theory of inside-out polytopes of Beck and Zaslavsky to prove structural results about the counting functions of Golomb rulers. We extend their approach to the various types of generalized Golomb rulers.

math.CO↗

arXiv2022

When is a numerical semigroup a quotient?

A natural operation on numerical semigroups is taking a quotient by a positive integer. If $\mathcal S$ is a quotient of a numerical semigroup with $k$ generators, we call $\mathcal S$ a $k$-quotient. We give a necessary condition for a given numerical semigroup $\mathcal S$ to be a $k$-quotient, and present, for each $k \ge 3$, the first known family of numerical semigroups that cannot be written as a $k$-quotient. We also examine the probability that a randomly selected numerical semigroup with $k$ generators is a $k$-quotient.

math.AC↗

arXiv2022

Bounds on Determinantal Complexity of Two Types of Generalized Permanents

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 permanents. One of these families is obtained by replacing permutations by signed permutations, and the other by replacing permutations by surjective functions with preimages of prescribed sizes.

math.CO↗

arXiv2021

Complex psd-minimal polytopes in dimensions two and three

The extension complexity of a polytope measures its amenability to succinct representations via lifts. There are several versions of extension complexity, including linear, real semidefinite, and complex semidefinite. We focus on the last of these, for which the least is known, and in particular on understanding which polytopes are complex psd-minimal. We prove the existence of an obstruction to complex psd-minimality which is efficiently computable via lattice membership problems. Using this tool, we complete the classification of complex psd-minimal polygons (geometrically as well as combinatorially). In dimension three we exhibit several new examples of complex psd-minimal polytopes and apply our obstruction to rule out many others.

math.CO↗

arXiv2020

A Plethora of Polynomials: A Toolbox for Counting Problems

A wide variety of problems in combinatorics and discrete optimization depend on counting the set $S$ of integer points in a polytope, or in some more general object constructed via discrete geometry and first-order logic. We take a tour through numerous problems of this type. In particular, we consider families of such sets $S_t$ depending on one or more integer parameters $t$, and analyze the behavior of the function $f(t)=|S_t|$. In the examples that we investigate, this function exhibits surprising polynomial-like behavior. We end with two broad theorems detailing settings where this polynomial-like behavior must hold. The plethora of examples illustrates the framework in which this behavior occurs and also gives an intuition for many of the proofs, helping us create a toolbox for counting problems like these.

math.CO↗

arXiv2020

A parametric version of LLL and some consequences: parametric shortest and closest vector problems

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: that is, they are given by formulas that are piecewise polynomial in t (for sufficiently large t), such that each piece is given by a congruence class modulo a period. As a consequence, we show that there are parametric solutions of the shortest vector problem (SVP) and closest vector problem (CVP) that are also eventually quasi-polynomial in t.

math.CO↗

arXiv2020

Constructing Partial MDS Codes from Reducible Curves

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 improved estimates on the smallest field sizes over which such codes can exist. Our results are obtained by combining ideas from projective algebraic geometry, combinatorics and probability theory.

cs.IT↗

arXiv2020

An Algebraic Approach to Projective Uniqueness with an Application to Order Polytopes

A combinatorial polytope $P$ is said to be projectively unique if it has a single realization up to projective transformations. Projective uniqueness is a geometrically compelling property but is difficult to verify. In this paper, we merge two approaches to projective uniqueness in the literature. One is primarily geometric and is due to McMullen, who showed that certain natural operations on polytopes preserve projective uniqueness. The other is more algebraic and is due to Gouveia, Macchia, Thomas, and Wiebe. They use certain ideals associated to a polytope to verify a property called graphicality that implies projective uniqueness. In this paper, we show that that McMullen's operations preserve not only projective uniquness but also graphicality. As an application, we show that large families of order polytopes are graphic and thus projectively unique.

math.CO↗

arXiv2019

Periodic behavior in families of numerical and affine semigroups via parametric Presburger arithmetic

Let $f_1(n), \ldots, f_k(n)$ be polynomial functions of $n$. For fixed $n\in\mathbb{N}$, let $S_n\subseteq \mathbb{N}$ be the numerical semigroup generated by $f_1(n),\ldots,f_k(n)$. As $n$ varies, we show that many invariants of $S_n$ are eventually quasi-polynomial in $n$, such as the Frobenius number, the type, the genus, and the size of the $Δ$-set. The tool we use is expressibility in the logical system of parametric Presburger arithmetic. Generalizing to higher dimensional families of semigroups, we also examine affine semigroups $S_n\subseteq \mathbb{N}^m$ generated be vectors whose coordinates are polynomial functions of $n$, and we prove similar results; for example, the Betti numbers are eventually quasi-polynomial functions of $n$.

math.CO↗

arXiv2018

An $\ell-p$ switch trick to obtain a new elementary proof of a criterion for arithmetic equivalence

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 almost every rational prime $\ell$ the arithmetic type of $\ell$ is the same in each field. In the 1990's Perlis and Stuart gave an unexpected characterization for arithmetic equivalence; they showed that to be arithmetically equivalent it is enough for almost every prime $\ell$ to have the same number of prime factors in each field. Here, using an $\ell-p$ switch trick, we provide an elementary proof of that fact based on a classical result of Smith from the 1870's.

math.NT↗

arXiv2018

Parametric Presburger Arithmetic: Complexity of Counting and Quantifier Elimination

We consider an expansion of Presburger arithmetic which allows multiplication by $k$ parameters $t_1,\ldots,t_k$. A formula in this language defines a parametric set $S_\mathbf{t} \subseteq \mathbb{Z}^{d}$ as $\mathbf{t}$ varies in $\mathbb{Z}^k$, and we examine the counting function $|S_\mathbf{t}|$ as a function of $\mathbf{t}$. For a single parameter, it is known that $|S_t|$ can be expressed as an eventual quasi-polynomial (there is a period $m$ such that, for sufficiently large $t$, the function is polynomial on each of the residue classes mod $m$). We show that such a nice expression is impossible with 2 or more parameters. Indeed (assuming \textbf{P} $\neq$ \textbf{NP}) we construct a parametric set $S_{t_1,t_2}$ such that $|S_{t_1, t_2}|$ is not even polynomial-time computable on input $(t_1,t_2)$. In contrast, for parametric sets $S_\mathbf{t} \subseteq \mathbb{Z}^d$ with arbitrarily many parameters, defined in a similar language without the ordering relation, we show that $|S_\mathbf{t}|$ is always polynomial-time computable in the size of $\mathbf{t}$, and in fact can be represented using the gcd and similar functions.

math.LO↗

arXiv2017

Parametric Presburger arithmetic: logic, combinatorics, and quasi-polynomial behavior

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 by t, and quantifiers on variables ranging over Z. That is, such families are defined using quantifiers and Boolean combinations of formulas of the form a(t) x <= b(t), where a(t) is in Z[t]^d, b(t) in Z[t]. A function g: N -> Z is a quasi-polynomial if there exists a period m and polynomials f_0, ..., f_{m-1} in Q[t] such that g(t)=f_i(t) for t congruent to i (mod m.) Recent results of Chen, Li, Sam; Calegari, Walker; Roune, Woods; and Shen concern specific families in parametric Presburger arithmetic that exhibit quasi-polynomial behavior. For example, S_t might be an a quasi-polynomial function of t or an element x(t) in S_t might be specifiable as a function with quasi-polynomial coordinates, for sufficiently large t. Woods conjectured that all parametric Presburger sets exhibit this quasi-polynomial behavior. Here, we prove this conjecture, using various tools from logic and combinatorics.

math.CO↗