Related papers: Visible lattice points along curves
{\it .}We completely characterize pairs of lattice points $P_1\neq P_2$ in the plane with the property that there are infinitely many lattice points $Q$ whose distance from both $P_1$ and $P_2$ is integral. In particular we show that it…
For a prime $p$ and an absolutely irreducible modulo $p$ polynomial $f(U,V) \in \Z[U,V]$ we obtain an asymptotic formulas for the number of solutions to the congruence $f(x,y) \equiv a \pmod p$ in positive integers $x \le X$, $y \le Y$,…
The purpose of this paper is to study low degree points on plane curves. We prove results analogous to those of Debarre and Klassen for singular plane curves with a finite number $\delta$ of ordinary nodes/cusps, where $\delta$ is bounded…
We introduce the concept of basis for a lattice. This basis plays a vital role to determine the completeness and consistency of the lattice. Weighted lattices are introduced and its complexity is formulated. Some axiomatic systems,…
An IC-plane graph is a topological graph where every edge is crossed at most once and no two crossed edges share a vertex. We show that every IC-plane graph has a visibility drawing where every vertex is an L-shape, and every edge is either…
One of the general problems in algebraic geometry is to determine algorithmically whether or not a given geometric object, defined by explicit polynomial equations (e.g. a curve or a surface), satisfies a given property (e.g. has…
We introduce one dimensional sets to help describe and constrain the integral curves of an $n$ dimensional dynamical system. These curves provide more information about the system than the zero-dimensional sets (fixed points) do. In fact,…
Asymptotics are given for the number of rational points in the domain of a morphism of weighted projective stacks whose images have bounded height and satisfy a (possibly infinite) set of local conditions. As a consequence we obtain results…
In this note is we exhibit an elementary method to construct explicitly curves over finite fields with many points. Despite its elementary character the method is very efficient and can be regarded as a partial substitute for the use of…
We consider planar curved strictly convex domains with no or very weak smoothness assumptions and prove sharp bounds for square-functions associated to the lattice point discrepancy.
We study equi-singular strata of plane curves with two singular points of prescribed types. The method of the previous work [Kerner06] is generalized to this case. In particular we consider the enumerative problem for plane curves with two…
We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques ([BP], [HBR]) and methods based on quasiorthogonality in the Mordell-Weil lattice…
A method is presented for calculating the Lie point symmetries of a scalar difference equation on a two-dimensional lattice. The symmetry transformations act on the equations and on the lattice. They take solutions into solutions and can be…
We study the collection of group structures that can be realized as a group of rational points on an elliptic curve over a finite field (such groups are well known to be of rank at most two). We also study various subsets of this collection…
We prove some new degeneracy results for integral points and entire curves on surfaces; in particular, we provide the first example, to our knowledge, of a simply connected smooth variety whose sets of integral points are never…
Integrable discrete scalar equations defined on a~two or a three dimensional lattice can be rewritten as difference systems in bond variables or in face variables respectively. Both the difference systems in bond variables and the…
The enumeration of normal surfaces is a key bottleneck in computational three-dimensional topology. The underlying procedure is the enumeration of admissible vertices of a high-dimensional polytope, where admissibility is a powerful but…
We provide an elementary proof of a formula for the number of northeast lattice paths that lie in a certain region of the plane. Equivalently, this formula counts the lattice points inside the Pitman--Stanley polytope of an n-tuple.
We will discuss a method for visual presentation of knotted surfaces in the four space, by examining a number and a position of its Morse's critical points. Using this method, we will investigate surface-knot with one critical point of…
We obtain a coarse relationship between geometric intersection numbers of curves and the sum of their subsurface projection distances with explicit quasi-constants. By using this relationship, we give applications in the studies of the…