arXiv subjects

Tristram Bogart

Publications and source records attributed to Tristram Bogart.

28 records · Page 2Linked to original sources

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↗

arXiv2015

A lower bound for the determinantal complexity of a hypersurface

We prove that the determinantal complexity of a hypersurface of degree $d > 2$ is bounded below by one more than the codimension of the singular locus, provided that this codimension is at least $5$. As a result, we obtain that the determinantal complexity of the $3 \times 3$ permanent is $7$. We also prove that for $n> 3$, there is no nonsingular hypersurface in $\mathbf{P}^n$ of degree $d$ that has an expression as a determinant of a $d \times d$ matrix of linear forms while on the other hand for $n \le 3$, a general determinantal expression is nonsingular. Finally, we answer a question of Ressayre by showing that the determinantal complexity of the unique (singular) cubic surface containing a single line is $5$.

cs.CC↗

arXiv2013

Few smooth d-polytopes with n lattice points

We prove that, for fixed n there exist only finitely many embeddings of Q-factorial toric varieties X into P^n that are induced by a complete linear system. The proof is based on a combinatorial result that for fixed nonnegative integers d and n, there are only finitely many smooth d-polytopes with n lattice points. We also enumerate all smooth 3-polytopes with at most 12 lattice points. In fact, it is sufficient to bound the singularities and the number of lattice points on edges to prove finiteness.

math.AG↗

arXiv2012

Hom-polytopes

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 category of vector spaces is limited, as we illustrate by a series of explicit examples exhibiting various extremal properties. The main challenge for hom-polytopes is to determine their vertices. A polytopal analogue of the rank-nullity theorem amounts to understanding how the vertex maps behave relative to their surjective and injective factors. This leads to interesting classes of surjective maps. In the last two sections we focus on two opposite extremal cases -- when the source and target polytopes are both polygons and are either generic or regular.

math.CO↗

arXiv2012

Obstructions to lifting tropical curves in surfaces in 3-space

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 curves in surfaces in 3-space. These are balanced rational weighted graphs in tropical surfaces. Specifically, we study the `lifting' problem: given a graph in a tropical surface, can one find a corresponding algebraic curve in a surface? We develop specific combinatorial obstructions to lifting a graph by reducing the problem to the question of whether or not one can factor a polynomial with particular support in the characteristic 0 case. This explains why some unusual tropical curves constructed by Vigeland are not liftable.

math.AG↗

arXiv2011

Enumeration of Golomb Rulers and Acyclic Orientations of Mixed Graphs

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 number theory in the 1930s and have attracted recent research activities on existence problems, such as the search for \emph{optimal} Golomb rulers (those of minimal length given a fixed number of markings). Our goal is to enumerate Golomb rulers in a systematic way: we study [g_m(t) := # {\x \in \Z^{m+1} : \, 0 = x_0 < x_1 < ... < x_{m-1} < x_m = t, \text{all} x_j - x_k \text{distinct}},] the number of Golomb rulers with $m+1$ markings and length $t$. Our main result is that $g_m(t)$ is a quasipolynomial in $t$ which satisfies a combinatorial reciprocity theorem: $(-1)^{m-1} g_m(-t)$ equals the number of rulers $\x$ of length $t$ with $m+1$ markings, each counted with its \emph{Golomb multiplicity}, which measures how many combinatorially different Golomb rulers are in a small neighborhood of $\x$. Our reciprocity theorem can be interpreted in terms of certain mixed graphs associated to Golomb rulers; in this language, it is reminiscent of Stanley's reciprocity theorem for chromatic polynomials. Thus in the second part of the paper we develop an analogue of Stanley's theorem to mixed graphs, which connects their chromatic polynomials to acyclic orientations.

math.CO↗

arXiv2010

Equality of Graver bases and universal Gröbner bases of colored partition identities

Associated to any vector configuration A is a toric ideal encoded by vectors in the kernel of A. Each toric ideal has two special generating sets: the universal Gröbner basis and the Graver basis. While the former is generally a proper subset of the latter, there are cases for which the two sets coincide. The most prominent examples among them are toric ideals of unimodular matrices. Equality of universal Gröbner basis and Graver basis is a combinatorial property of the toric ideal (or, of the defining matrix), providing interesting information about ideals of higher Lawrence liftings of a matrix. Nonetheless, a general classification of all matrices for which both sets agree is far from known. We contribute to this task by identifying all cases with equality within two families of matrices; namely, those defining rational normal scrolls and those encoding homogeneous primitive colored partition identities.

math.AC↗

arXiv2009

Small Chvatal rank

We propose a variant of the Chvatal-Gomory procedure that will produce a sufficient set of facet normals for the integer hulls of all polyhedra {xx : Ax <= b} as b varies. The number of steps needed is called the small Chvatal rank (SCR) of A. We characterize matrices for which SCR is zero via the notion of supernormality which generalizes unimodularity. SCR is studied in the context of the stable set problem in a graph, and we show that many of the well-known facet normals of the stable set polytope appear in at most two rounds of our procedure. Our results reveal a uniform hypercyclic structure behind the normals of many complicated facet inequalities in the literature for the stable set polytope. Lower bounds for SCR are derived both in general and for polytopes in the unit cube.

math.OC↗

arXiv2005

The Circuit Ideal of a Vector Configuration

The circuit ideal, $\ica$, of a configuration $\A = \{\a_1, ..., \a_n\} \subset \Z^d$ is the ideal generated by the binomials ${\x}^{\cc^+} - {\x}^{\cc^-} \in \k[x_1, ..., x_n]$ as $\cc = \cc^+ - \cc^- \in \Z^n$ varies over the circuits of $\A$. This ideal is contained in the toric ideal, $\ia$, of $\A$ which has numerous applications and is nontrivial to compute. Since circuits can be computed using linear algebra and the two ideals often coincide, it is worthwhile to understand when equality occurs. In this paper we study $\ica$ in relation to $\ia$ from various algebraic and combinatorial perspectives. We prove that the obstruction to equality of the ideals is the existence of certain polytopes. This result is based on a complete characterization of the standard pairs/associated primes of a monomial initial ideal of $\ica$ and their differences from those for the corresponding toric initial ideal. Eisenbud and Sturmfels proved that $\ia$ is the unique minimal prime of $\ica$ and that the embedded primes of $\ica$ are indexed by certain faces of the cone spanned by $\A$. We provide a necessary condition for a particular face to index an embedded prime and a partial converse. Finally, we compare various polyhedral fans associated to $\ia$ and $\ica$. The Gröbner fan of $\ica$ is shown to refine that of $\ia$ when the codimension of the ideals is at most two.

math.AC↗

arXiv2005

Computing Tropical Varieties

The tropical variety of a $d$-dimensional prime ideal in a polynomial ring with complex coefficients is a pure $d$-dimensional polyhedral fan. This fan is shown to be connected in codimension one. We present algorithmic tools for computing the tropical variety, and we discuss our implementation of these tools in the Gröbner fan software \texttt{Gfan}. Every ideal is shown to have a finite tropical basis, and a sharp lower bound is given for the size of a tropical basis for an ideal of linear forms.

math.AG↗