Related papers: Euclid meets Bezout: Intersecting algebraic plane …
We study the structure of collections of algebraic curves in three dimensions that have many curve-curve incidences. In particular, let $k$ be a field and let $\mathcal{L}$ be a collection of $n$ space curves in $k^3$, with…
Given two distinct reduced, irreducible curves of given degrees, contained in projective space but whose union is not contained in a hyperplane, what is the largest number of points of intersection they can have? When the projective space…
This paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are…
This paper focuses on intersection of closed curves on translation surfaces. Namely, we investigate the question of determining the intersection of two closed curves of a given length on such surfaces. This question has been investigated in…
We present a complete classification of complex plane algebraic curves, equipped with the induced Euclidean, up to global bilipschitz homeomorphism.
We determine the crossing number of polynomial size curve systems on standard surfaces, in terms of the genus, up to high precision.
The purpose of this paper is twofold. An immediate practical use of the presented algorithm is its applicability to the parametric solution of underdetermined linear ordinary differential equations (ODEs) with coefficients that are…
We present a new certified and complete algorithm to compute arrangements of real planar algebraic curves. Our algorithm provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic…
We provide an algorithm to check whether two rational space curves are related by a similarity. The algorithm exploits the relationship between the curvatures and torsions of two similar curves, which is formulated in a computer algebra…
The article presents a new approach to euclidean plane geometry based on projective geometric algebra (PGA). It is designed for anyone with an interest in plane geometry, or who wishes to familiarize themselves with PGA. After a brief…
We present a certified algorithm based on subdivision for computing an isotopic approximation to any number of curves in the plane. Our algorithm is based on the certified curve approximation algorithm of Plantinga and Vegter. The main…
We present a certified algorithm based on subdivision for computing an isotopic approximation to any number of curves in the plane. Our algorithm is based on the certified curve approximation algorithm of Plantinga and Vegter. The main…
The classical version of B\'ezout's Theorem gives an integer-valued count of the intersection points of hypersurfaces in projective space over an algebraically closed field. Using work of Kass and Wickelgren, we prove a version of…
We enumerate plane complex algebraic curves of a given degree with one singularity of any given topological type. Our approach is to compute the homology classes of the corresponding equisingular strata in the parameter spaces of plane…
An algorithm for generating interpolants for formulas which are conjunctions of quadratic polynomial inequalities (both strict and nonstrict) is proposed. The algorithm is based on a key observation that quadratic polynomial inequalities…
We present novel, deterministic, efficient algorithms to compute the symmetries of a planar algebraic curve, implicitly defined, and to check whether or not two given implicit planar algebraic curves are similar, i.e. equal up to a…
The problem of finding a point in the intersection of closed sets can be solved by the method of alternating projections and its variants. It was shown in earlier papers that for convex sets, the strategy of using quadratic programming (QP)…
This paper improves the algorithms based on supporting halfspaces and quadratic programming for convex set intersection problems in our earlier paper in several directions. First, we give conditions so that much smaller quadratic programs…
We produce algorithms to detect whether a complex affine variety computed and presented numerically by the machinery of numerical algebraic geometry corresponds to an associated component of a polynomial ideal.
We study how the supporting hyperplanes produced by the projection process can complement the method of alternating projections and its variants for the convex set intersection problem. For the problem of finding the closest point in the…