Related papers: Beyond the Richter-Thomassen Conjecture
We study the following question: given a set P of 3d-2 points and an immersed curve G in the real plane R^2, all in general position, how many real rational plane curves of degree d pass through these points and are tangent to this curve.…
We consider the non-crossing connectors problem, which is stated as follows: Given n simply connected regions R_1,...,R_n in the plane and finite point sets P_i subset of R_i for i=1,...,n, are there non-crossing connectors y_i for…
For each integer $k \in [0,9]$, we count the number of plane cubic curves defined over a finite field $\mathbb{F}_q$ that do not share a common component and intersect in exactly $k\ \mathbb{F}_q$-rational points. We set this up as a…
A curve \gamma in the plane is t-monotone if its interior has at most t-1 vertical tangent points. A family of t-monotone curves F is \emph{simple} if any two members intersect at most once. It is shown that if F is a simple family of n…
Consider a random geometric graph $G$ with a vertex set defined by a Poisson point process with intensity $t>0$ in a convex body. We can generate a drawing of the graph by projecting the construction onto some plane $L$. Choosing different…
A closed plane meander of order n is a closed self-avoiding loop intersecting an infinite line 2n times. Meanders are considered distinct up to any smooth deformation leaving the line fixed. We have developed an improved algorithm, based on…
The matching problem for a given Jordan curve in the complex plane asks to find two nonconstant functions, one analytic in the bounded complementary component of the curve and the other analytic in the unbounded complementary component of…
Simple drawings are drawings of graphs in the plane or on the sphere such that vertices are distinct points, edges are Jordan arcs connecting their endpoints, and edges intersect at most once (either in a proper crossing or in a shared…
Two families $\mathcal{A}$ and $\mathcal{B}$ of sets are said to be cross-intersecting if each member of $\mathcal{A}$ intersects each member of $\mathcal{B}$. For any two integers $n$ and $k$ with $0 \leq k \leq n$, let ${[n] \choose \leq…
Given $n$ points in the plane, a \emph{covering path} is a polygonal path that visits all the points. If no three points are collinear, every covering path requires at least $n/2$ segments, and $n-1$ straight line segments obviously suffice…
We discuss a special case of a family of means defined using intersections of osculating hyperplanes to curves in R^n. Let C be the curve in R^n with vector equation x_{k}(t)=t(ln t)^{k-1},k=1,...,n. Let O_{k} be the osculating hyperplane…
Consider $n$ points evenly spaced on a circle, and a path of $n-1$ chords that uses each point once. There are $m=\lfloor n/2\rfloor$ possible chord lengths, so the path defines a multiset of $n-1$ elements drawn from $\{1,2,\ldots,m\}$.…
The computation of the topology of a real algebraic plane curve is greatly simplified if there are no more than one critical point in each vertical line: the general position condition. When this condition is not satisfied, then a finite…
Given an ordered sequence of $N$-choose-2 integers, we give necessary and sufficient conditions to have an ordered collection of $N$ simple closed curves on a torus such that the algebraic pairwise intersections of those curves are the…
Let $S$ be a set of $n$ points in general position in the plane. Join every pair of points in $S$ with a straight line segment. Let $\overline{cr}(S)$ be number of pairs of these edges that intersect in their interior. Suppose that this…
We prove a symmetric version of B\'ezout's theorem. More precisely, we show that the symmetric orbit type of a transverse intersection of complex symmetric hypersurfaces in projective space is determined by the degrees. In the projective…
Let $\F$ be a family of $n$ pairwise intersecting circles in the plane. We show that the number of lenses, that is convex digons, in the arrangement induced by $\F$ is at most $2n-2$. This bound is tight. Furthermore, if no two circles in…
In this paper, we investigate the problem of separating a set $X$ of points in $\mathbb{R}^{2}$ with an arrangement of $K$ lines such that each cell contains an asymptotically equal number of points (up to a constant ratio). We consider a…
We show that the maximum number of pairwise non-overlapping $k$-rich lenses (lenses formed by at least $k$ circles) in an arrangement of $n$ circles in the plane is $O\left(\frac{n^{3/2}\log{(n/k^3)}}{k^{5/2}} + \frac{n}{k} \right)$, and…
Given a set of $n$ points in $R^2$, the Szemer\'edi-Trotter theorem establishes that the number of lines which can be incident to at least $k > 1$ of these points is $O(n^2/k^3 + n/k)$. J.\ Solymosi conjectured that if one requires the…