Related papers: On the Twelve-Point Theorem for $\ell$-Reflexive P…

The Ehrhart polynomial of a lattice polygon P is completely determined by the pair (b(P),i(P)) where b(P) equals the number of lattice points on the boundary and i(P) equals the number of interior lattice points. All possible pairs…

We use the finite lattice method to count the number of punctured staircase and self-avoiding polygons with up to three holes on the square lattice. New or radically extended series have been derived for both the perimeter and area…

Let $A$ be an abelian surface over $\mathbb{Q}$ whose geometric endomorphism ring is a maximal order in a non-split quaternion algebra. Inspired by Mazur's theorem for elliptic curves, we show that the torsion subgroup of $A(\mathbb{Q})$ is…

A reflexive cycle is any reflexive digraph whose underlying undirected graph is a cycle. Call a relational structure Slupecki if its surjective polymorphisms are all essentially unary. We prove that all reflexive cycles of girth at least 4…

An elliptic pair $(X, C)$ is a generalization of a rational elliptic fibration $X \to \mathbb{P}^1$ with fiber $C,$ introduced in \cite{jenia_blowup}. Here, $X$ is a projective rational surface with log terminal singularities, and $C$ is an…

We prove that there are at most 13 real quadratic fields that admit a ternary universal quadratic lattice, thus establishing a strong version of Kitaoka's Conjecture for quadratic fields. More generally, we obtain explicit upper bounds on…

The classical Steinitz theorem asserts that if the origin lies within the interior of the convex hull of a set $S \subset \mathbb{R}^d$, then there are at most $2d$ points in $S$ whose convex hull contains the origin within its interior.…

In Part I we gave a polynomial growth lower-bound for the number of nodal domains of a Hecke-Maass cuspform in a compact part of the modular surface, assuming a Lindel\"of hypothesis. That was a consequence of a topological argument and…

We prove the existence of infinitely many solutions to an elliptic problem by borrowing the techniques from algebraic topology. The solution(s) thus obtained will also be proved to be bounded.

For a large class of isotrivial rational elliptic surfaces (with section), we show that the set of rational points is dense for the Zariski topology, by carefully studying variations of root numbers among the fibers of these surfaces. We…

We show that there are infinitely many elliptic curves $E/\mathbb{Q}$, up to isomorphism over $\overline{\mathbb{Q}}$, for which the finitely generated group $E(\mathbb{Q})$ has rank exactly $2$. Our elliptic curves are given by explicit…

We show that there are 4 infinite families of lattice equable kites, given by corresponding Pell or Pell-like equations, but up to Euclidean motions, there are exactly 5 lattice equable trapezoids (2 isosceles, 2 right, 1 singular) and 4…

We classify del Pezzo surfaces with 1/3(1,1) points in 29 qG-deformation families grouped into six unprojection cascades (this overlaps with work of Fujita and Yasutake), we tabulate their biregular invariants, we give good model…

We show the density of rational points on non-isotrivial elliptic surfaces by studying the variation of the root numbers among the fibers of these surfaces, conditionally to two analytic number theory conjectures (the squarefree conjecture…

Hexagonal circle patterns are introduced, and a subclass thereof is studied in detail. It is characterized by the following property: For every circle the multi-ratio of its six intersection points with neighboring circles is equal to -1.…

For any lattice polytope $P$, we consider an associated polynomial $\bar{\delta}_{P}(t)$ and describe its decomposition into a sum of two polynomials satisfying certain symmetry conditions. As a consequence, we improve upon known…

We present the last missing details of our algorithm for the classification of reflexive polyhedra in arbitrary dimensions. We also present the results of an application of this algorithm to the case of three dimensional reflexive…

Two polygons are amicable if the perimeter of one is equal to the area of the other and vice versa. A polygon is a lattice polygon if its vertices are on the integer lattice $\Z^2$. We show that there is one pair of amicable lattice…

Reflexive polytopes form one of the distinguished classes of lattice polytopes. Especially reflexive polytopes which possess the integer decomposition property are of interest. In the present paper, by virtue of the algebraic technique on…

We study upper bounds on the number of lattice points for convex bodies having their centroid at the origin. For the family of simplices as well as in the planar case we obtain best possible results. For arbitrary convex bodies we provide…