Related papers: Intersection numbers in the curve complex via subs…
The characteristic numbers of smooth plane quartics are computed using intersection theory on a component of the moduli space of stable maps. This completes the verification of Zeuthen's prediction of characteristic numbers of smooth plane…
By studying $\mathbb{A}^1$-curves on varieties, we propose a geometric approach to strong approximation problem over function fields of complex curves. We prove that strong approximation holds for smooth, low degree affine complete…
We present a new method for visualizing implicit real algebraic curves inside a bounding box in the $2$-D or $3$-D ambient space based on numerical continuation and critical point methods. The underlying techniques work also for tracing…
We estimate the maximal number of integral points which can be on a convex arc in the plane with given length, minimal radius of curvature and initial slope.
By a curve in R^d we mean a continuous map gamma:I -> R^d, where I is a closed interval. We call a curve gamma in R^d at most k crossing if it intersects every hyperplane at most k times (counted with multiplicity). The at most d crossing…
A tropical complete intersection curve C in R^(n+1) is a transversal intersection of n smooth tropical hypersurfaces. We give a formula for the number of vertices of C given by the degrees of the tropical hypersurfaces. We also compute the…
We propose sublinear algorithms for probabilistic testing of the discrete and continuous Fr\'echet distance - a standard similarity measure for curves. We assume the algorithm is given access to the input curves via a query oracle: a query…
A novel and deterministic algorithm is presented to detect whether two given rational plane curves are related by means of a similarity, which is a central question in Pattern Recognition. As a by-product it finds all such similarities, and…
If two closed Jordan curves in the plane have precisely one point in common, then it is called a {\em touching point}. All other intersection points are called {\em crossing points}. The main result of this paper is a Crossing Lemma for…
We show that the detection of geometric intersection in an arbitrary representation of the mapping class group of surface implies the injectivity of that representation up to center, and vice versa. As an application, we discuss the…
We prove that the length difference between a closed periodic curve and its parallel curve at a sufficiently small distance is proportional to the rotation index. As an application, the rotation index of a curve could be estimated by means…
We present an algorithm for computing curves and families of curves of prescribed degree and geometric genus on real rational surfaces.
Properties of a parametric curve in R^3 are often determined by analysis of its piecewise linear (PL) approximation. For Bezier curves, there are standard algorithms, known as subdivision, that recursively create PL curves that converge to…
We provide the first non-trivial examples of quasi-isometric embeddings between curve complexes. These are induced either by puncturing a closed surface or via orbifold coverings. As a corollary, we give new quasi-isometric embeddings…
In this article, we study isomorphisms between complements of irreducible curves in the projective plane $\mathbb{P}^2$, over an arbitrary algebraically closed field. Of particular interest are rational unicuspidal curves. We prove that if…
We find bounds on the difference between the writhing number of a smooth curve, and the writhing number of a polygon inscribed within. The proof is based on an extension of Fuller's difference of writhe formula to the case of polygonal…
In recent years, networks with higher-order interactions have emerged as a powerful tool to model complex systems. Comparing these higher-order systems remains however a challenge. Traditional similarity measures designed for pairwise…
Geometry and topology have generated impacts far beyond their pure mathematical primitive, providing a solid foundation for many applicable tools. Typically, real-world data are represented as vectors, forming a linear subspace for a given…
Kontsevich's work on Airy matrix integrals has led to explicit results for the intersection numbers of the moduli space of curves. In this article we show that a duality between k-point functions on $N\times N$ matrices and N-point…
Near a singular point of a surface or a curve, geometric invariants diverge in general, and the orders of diverge, in particular the boundedness about these invariants represent geometry of the surface and the curve. In this paper, we study…