Related papers: On Equal Point Separation by Planar Cell Decomposi…

The paper is concerned with families of plane algebraic curves that contain a given and quite special finite set X of points in the projective plane. We focus on the case in which the set X is formed by transversally intersecting pairs of…

We classify elliptic curves over the rationals whose N\'eron model over the integers is semi-abelian, with good reduction at p=2, and whose Mordell--Weil group contains an element of order two that stays non-trivial at p=2. Furthermore, we…

Let $X\subseteq \mathbb{P}^3$ be a smooth projective surface of degree $d\ge 4$ defined over a number field $K$, and let $N_{X^{\prime}}(B)$ be the number of rational points of $X$ of height at most $B$ that do not lie on lines contained in…

This paper investigates linear programming based branch-and-bound using general disjunctions, also known as stabbing planes, for solving integer programs. We derive the first sub-exponential lower bound (in the encoding length $L$ of the…

Let $X$ be an irreducible projective variety of dimension $n$ in a projective space and let $x$ be a point of $X$. Denote by ${\rm Curves}_d(X,x)$ the space of curves of degree $d$ lying on $X$ and passing through $x$. We will show that the…

Let $G$ be a connected, simply connected, simple, complex, linear algebraic group. Let $P$ be an arbitrary parabolic subgroup of $G$. Let $X=G/P$ be the $G$-homogeneous projective space attached to this situation. We consider the (small)…

It is known that every proper minor-closed class of graphs has bounded stack-number (a.k.a. book thickness and page number). While this includes notable graph families such as planar graphs and graphs of bounded genus, many other graph…

Let E be an elliptic curve over Q, and let n=>1. The central object of study of this article is the division field Q(E[n]) that results by adjoining to Q the coordinates of all n-torsion points on E(Q). In particular, we classify all curves…

Given a smooth complex variety $X$, an algebraically skew embedding of $X$ is an embedding of $X$ into a complex projective space $\mathbb{P}^N$ such that for any two points $x,y\in X$, their embedded tangent spaces in $\mathbb{P}^N$ do not…

We reformulate a fundamental result due to Cook, Harbourne, Migliore and Nagel on the existence and irreduciblity of unexpected plane curves of a set of points $Z$ in $\mathbb{P}^2$, using the minimal degree of a Jacobian syzygy of the…

Let $\{p_1, \ldots , p_n \} \subset {\Bbb{R}}^2$ be a separated point set, i.e., any two points have a distance at least $1$. Let $k \ge 1$ be an integer, and $1 \le t_1 < \ldots < t_k$ be real numbers. Let $\delta > 0$. Suppose for all $1…

Given a set $S$ of $n$ line segments in the plane, we say that a region $\mathcal{R}\subseteq \mathbb{R}^2$ is a {\em stabber} for $S$ if $\mathcal{R}$ contains exactly one endpoint of each segment of $S$. In this paper we provide optimal…

We study a class of geometric covering and packing problems for bounded regions on the plane. We are given a set of axis-parallel line segments that induces a planar subdivision with a set of bounded (rectilinear) faces. We are interested…

We investigate combinatorial bounds for the total Tjurina numbers of plane curve arrangements. Focusing on arrangements of lines and conics in $\mathbb{P}^2$ that admit only ordinary quasi-homogeneous singularities, we derive new structural…

A smooth real curve is called separating in case the complement of the real locus inside the complex locus is disconnected. This is the case if there exists a morphism to the projective line whose inverse image of the real locus of the…

The scattering of waves by obstacles in a 2D setting is considered, in particular the computation of the scattered field via the collocation or the least-squares methods. In the case of multiple scattering by smooth obstacles, we prove that…

Let $K$ be a convex body in Euclidean space ${\mathbb R}^d$, and let a translation invariant, locally finite Borel measure on the space of hyperplanes in ${\mathbb R}^d$ be given. For $\delta\ge 0$, we consider the set of all points $x$ for…

Here is present short proofing of Jordan's theorem about dividing of flat on two disjoint subsets by one closed curve.

We give a new perspective of Heegaard splittings in terms square complexes and Guirardel's notion of a \textit{core} which allows for combinatorial measurement of the obstruction to being a connect sum of Heegaard diagrams. A Heegaard…

Given a finite $n$-element set $X$, a family of subsets ${\mathcal F}\subset 2^X$ is said to separate $X$ if any two elements of $X$ are separated by at least one member of $\mathcal F$. It is shown that if $|\mathcal F|>2^{n-1}$, then one…