Related papers: On Lattice Diameter Segments and A Discrete Borsuk…
We present a polynomial partitioning theorem for finite sets of points in the real locus of an irreducible complex algebraic variety of codimension at most two. This result generalizes the polynomial partitioning theorem on the Euclidean…
In 2015, Guth proved that if $S$ is a collection of $n$ $g$-dimensional semi-algebraic sets in $\mathbb{R}^d$ and if $D\geq 1$ is an integer, then there is a $d$-variate polynomial $P$ of degree at most $D$ so that each connected component…
The combinatorial properties of partitions with various restrictions on their hooksets are explored. A connection with numerical semigroups extends current results on simultaneous s/t-cores. Conditions that suffice for a partition to…
Spectra of suitably chosen Pisot-Vijayaraghavan numbers represent non-trivial examples of self-similar Delone point sets of finite local complexity, indispensable in quasicrystal modeling. For the case of quadratic Pisot units we…
We begin by revisiting a paper of Erd\H{o}s and Fishburn, which posed the following question: given $k\in \mathbb{N}$, what is the maximum number of points in a plane that determine at most $k$ distinct distances, and can such optimal…
We study error-correcting codes in the space $\mathcal{S}_{n,q}$ of length-$n$ multisets over a $q$-ary alphabet under the deletion metric, motivated by permutation channels in which ordering is completely lost and errors act only on symbol…
A seminal result of E. Ehrhart states that the number of integer lattice points in the dilation of a rational polytope by a positive integer $k$ is a quasi-polynomial function of $k$ --- that is, a "polynomial" in which the coefficients are…
We consider approximation of diameter of a set $S$ of $n$ points in dimension $m$. E$\tilde{g}$ecio$\tilde{g}$lu and Kalantari \cite{kal} have shown that given any $p \in S$, by computing its farthest in $S$, say $q$, and in turn the…
We classify, according to their computational complexity, integer optimization problems whose constraints and objective functions are polynomials with integer coefficients and the number of variables is fixed. For the optimization of an…
There are two well known types of algorithms for solving CSPs: local propagation and generating a basis of the solution space. For several years the focus of the CSP research has been on `hybrid' algorithms that somehow combine the two…
A lattice point in $\mathbb{R}^2$ is a point $(x,y)$ with $x,y\in\mathbb{Z}$, and a lattice triangle is a triangle whose three vertices are all lattice points. We investigate the integers $k$ with the property that if $T$ is a lattice…
We study ways to expedite Yates's algorithm for computing the zeta and Moebius transforms of a function defined on the subset lattice. We develop a trimmed variant of Moebius inversion that proceeds point by point, finishing the calculation…
The Semialgebraic Orbit Problem is a fundamental reachability question that arises in the analysis of discrete-time linear dynamical systems such as automata, Markov chains, recurrence sequences, and linear while loops. An instance of the…
We consider the problem of revealing a small hidden lattice from the knowledge of a low-rank sublattice modulo a given sufficiently large integer -- the {\em Hidden Lattice Problem}. A central motivation of study for this problem is the…
We study the problem of partitioning a polygon into the minimum number of subpolygons using cuts in predetermined directions such that each resulting subpolygon satisfies a given width constraint. A polygon satisfies the unit-width…
We characterize numerical semigroups for which the poset of its ideal class monoid is a lattice, and study the irreducible elements of such a lattice with respect to union, intersection, infimum and supremum.
Given any full rank lattice and a natural number N , we regard the point set given by the scaled lattice intersected with the unit square under the Lambert map to the unit sphere, and show that its spherical cap discrepancy is at most of…
Join-distributive lattices are finite, meet-semidistributive, and semimodular lattices. They are the same as Dilworth's lattices in 1940, and many alternative definitions and equivalent concepts have been discovered or rediscovered since…
Polynomial reconstruction on Cartesian grids is fundamental in many scientific and engineering applications, yet it is still an open problem how to construct for a finite subset $K$ of $\mathbb{Z}^{\textsf{D}}$ a lattice $\mathcal{T}\subset…
Since Henrik Strietz's 1975 paper proving that the lattice Part($n$) of all partitions of an $n$-element finite set is four-generated, more than half a dozen papers have been devoted to four-element generating sets of this lattice. We prove…