On Lattice Barycentric Tetrahedra

We say a lattice tetrahedron whose centroid is its only non-vertex lattice point is lattice barycentric. The notation T(a,b,c) describes the lattice tetrahedron with vertices {0, e_1, e_2, a e_1 + b e_2 + c e_3}. Our result is that all such T(a,b,c) are unimodularly equivalent to T(3,3,4) or T(7,11,20).Comment: 9 pages, 4 table

Lattice polytopes with distinct pair-sums

Let P be a lattice polytope in R^n, and let P \cap Z^n = {v_1,...,v_N}. If the N + \binom N2 points 2v_1,...,2v_N; v_1+v_2,...v_{N-1}+v_N are distinct, we say that P is a "distinct pair-sum" or "dps" polytope. We show that, if P is a dsp polytope in R^n, then N \le 2^n, and, for every n, we construct dps polytopes in R^n which contain 2^n lattice points. We also discuss the relation between dps polytopes and the study of sums of squares of real polynomials.Comment: 8 pages. Submitted to the Special Issue on Geometric Combinatorics of the journal "Discrete and Computational Geometry

A note on mediated simplices

Many homogeneous polynomials that arise in the study of sums of squares and Hilbert's 17th problem come from monomial substitutions into the arithmetic-geometric inequality. In 1989, the second author gave a necessary and sufficient condition for such a form to have a representation as a sum of squares of forms (Math. Ann., (283), 431--464), involving the arrangement of lattice points in the simplex whose vertices were the $n$-tuples of the exponents used in the substitution. Further, a claim was made, and not proven, that sufficiently large dilations of any such simplex will also satisfy this condition. The aim of this short note is to prove the claim, and provide further context for the result, both in the study of Hilbert's 17th Problem and the study of lattice point simplices.Comment: Submitted to the Proceedings of the 2019 Arctic Applied Algebra conference in Troms{\o}, Norwa

Classifying terminal weighted projective space

We present a classification of all weighted projective spaces with at worst terminal or canonical singularities in dimension four. As a corollary we also classify all four-dimensional one-point lattice simplices up to equivalence. Finally, we classify the terminal Gorenstein weighted projective spaces up to dimension ten.Comment: 19 pages, 6 table

On the span of lattice points in a parallelepiped

Let $\Lambda\subset\mathbf{R}^{n}$ be a lattice which contains the integer lattice $\mathbf{Z}^{n}$. We characterize the space of linear functions $\mathbf{R}^{n}\rightarrow\mathbf{R}$ which vanish on the lattice points of $\Lambda$ lying in the half-open unit cube $[0,1)^{n}$. We also find an explicit formula for the dimension of the linear span of $\Lambda\cap[0,1)^{n}$. The results in this paper generalize and are based on the Terminal Lemma of Reid, which is in turn based upon earlier work of Morrison and Stevens on the classification of four dimensional isolated Gorenstein terminal cyclic quotient singularities.Comment: 26 page

Minimal volume kk-point lattice dd-simplices

We extend the results of Bey, Hen, and Wills (http://arxiv.org/abs/math/0606089). In this paper, we show that, up to equivalence under unimodular transformations, there is exactly one class of $d$-simplices having $k \ge 1$ interior lattice points and minimal volume $\frac{1}{d!}(dk+1)$.Comment: 22 pages, 6 figure

Clean Lattice Tetrahedra

A clean lattice tetrahedron is a non-degenerate tetrahedron with the property that the only lattice points on its boundary are its vertices. We present some new proofs of old results and some new results on clean lattice tetrahedra, with an emphasis on counting the number of its interior lattice points and on computing its lattice width.Comment: submitted for publicatio

Coprime Ehrhart theory and counting free segments

A lattice polytope is "free" (or "empty") if its vertices are the only lattice points it contains. In the context of valuation theory, Klain (1999) proposed to study the functions $\alpha_i(P;n)$ that count the number of free polytopes in $nP$ with $i$ vertices. For $i=1$, this is the famous Ehrhart polynomial. For $i > 3$, the computation is likely impossible and for $i=2,3$ computationally challenging. In this paper, we develop a theory of coprime Ehrhart functions, that count lattice points with relatively prime coordinates, and use it to compute $\alpha_2(P;n)$ for unimodular simplices. We show that the coprime Ehrhart function can be explicitly determined from the Ehrhart polynomial and we give some applications to combinatorial counting.Comment: v2: 8 pages, minor additions, accepted for publication in International Mathematics Research Notice

Amoebas of genus at most one

The amoeba of a Laurent polynomial f \in \C[z_1^{\pm 1},\ldots,z_n^{\pm 1}] is the image of its zero set $\mathcal{V}(f)$ under the log-absolute-value map. Understanding the space of amoebas (i.e., the decomposition of the space of all polynomials, say, with given support or Newton polytope, with regard to the existing complement components) is a widely open problem. In this paper we investigate the class of polynomials $f$ whose Newton polytope \New(f) is a simplex and whose support $A$ contains exactly one point in the interior of \New(f). Amoebas of polynomials in this class may have at most one bounded complement component. We provide various results on the space of these amoebas. In particular, we give upper and lower bounds in terms of the coefficients of $f$ for the existence of this complement component and show that the upper bound becomes sharp under some extremal condition. We establish connections from our bounds to Purbhoo's lopsidedness criterion and to the theory of $A$-discriminants. Finally, we provide a complete classification of the space of amoebas for the case that the exponent of the inner monomial is the barycenter of the simplex Newton polytope. In particular, we show that the set of all polynomials with amoebas of genus 1 is path-connected in the corresponding space of amoebas, which proves a special case of the question on connectivity (for general Newton polytopes) stated by H. Rullg{\aa}rd.Comment: 26 pages, 5 figures; minor revisio

A lower bound technique for triangulations of simplotopes

Products of simplices, called simplotopes, and their triangulations arise naturally in algorithmic applications in game theory and optimization. We develop techniques to derive lower bounds for the size of simplicial covers and triangulations of simplotopes, including those with interior vertices. We establish that a minimal triangulation of a product of two simplices is given by a vertex triangulation, i.e., one without interior vertices. For products of more than two simplices, we produce bounds for products of segments and triangles. Aside from cubes, these are the first known lower bounds for triangulations of simplotopes with three or more factors, and our techniques suggest extensions to products of other kinds of simplices. We also construct a minimal triangulation of size 10 for the product of a triangle and a square using our lower bound.Comment: 31 pages, related work at http://www.math.hmc.edu/~su/papers.htm