Related papers: Division Polynomials and Intersection of Projectiv…
We construct pairs of elliptic curves over number fields with large intersection of projective torsion points.
We study effective versions of unlikely intersections of images of torsion points of elliptic curves on the projective line.
We investigate the duality between local (complex analytic) projective structures on surfaces and two dimensional (complex analytic) neighborhoods of rational curves having self-intersection +1. We study the analytic classification,…
We describe methods to determine all the possible torsion groups of an elliptic curve that actually appear over a fixed quadratic field. We use these methods to find, for each group that can appear over a quadratic field, the field with the…
This paper is devoted to presenting a new approach to determine the intersection of two quadrics based on the detailed analysis of its projection in the plane (the so called cutcurve) allowing to perform the corresponding lifting correctly.…
We study intersections of projective convex sets in the sense of Steinitz. In a projective space, an intersection of a nonempty family of convex sets splits into multiple connected components each of which is a convex set. Hence, such an…
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…
We consider the popular and classical method of alternating projections for finding a point in the intersection of two closed sets. By situating the algorithm in a metric space, equipped only with well-behaved geodesics and angles (in the…
We study algebraic and combinatorial aspects of (classical) projections of $m$-dimensional tropical varieties onto $(m+1)$-dimensional planes. Building upon the work of Sturmfels, Tevelev, and Yu on tropical elimination as well as the work…
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…
The average value of the number of intersection points of two plane curves given by trigonometrical polynomial maps of degree $N$ and bounded $L_2$- or $W_2^r$- norms is calculated
Suppose E_1, E_2 are elliptic curves (over the complex numbers) together with standard double coverings of the projective line identifying a point and its inverse on E_i. Bogomolov, Fu and Tschinkel have asked if the number of common images…
In this paper, we compute the number of self-intersections of a plane projection of a generic complete intersection curve defined by polynomials with the given support. Moreover, we discuss the tropical counterpart of this problem.
We show how the Eulcidean algorithm for polynomials can be used to find the intersection points, with multiplicities, of two plane algebraic curves.
This paper presents division polynomials for twisted Edwards curves. Their chief property is that they characterise the $n$-torsion points of a given twisted Edwards curve. We also present results concerning the coefficients of these…
We establish a connection between torsion packets on curves of genus $2$ and pairs of elliptic curves realized as double covers of the projective line $\mathbb{P}_{x}^{1}$ that have many common torsion $x$-coordinates. This can be used to…
We consider the method of alternating projections for finding a point in the intersection of two closed sets, possibly nonconvex. Assuming only the standard transversality condition (or a weaker version thereof), we prove local linear…
The arithmetic of elliptic curves, namely polynomial addition and scalar multiplication, can be described in terms of global sections of line bundles on $E\times E$ and $E$, respectively, with respect to a given projective embedding of $E$…
Obtaining complete information about the shape of an object by looking at it from a single direction is impossible in general. In this paper, we theoretically study obtaining differential geometric information of an object from orthogonal…
This paper presents two kinds of division polynomials for twisted Edwards curves. Their chief property is that they characterise the $n$-torsion points of a given twisted Edwards curve. We also present results concerning the coefficients of…