We provide a new technique to detect the singularities of rational space curves. Given a rational parametrization of a space curve, we first compute a μ-basis for the parametrization. From this μ-basis we generate three planar algebraic curves of different bidegrees whose intersection points correspond to the parameters of the singularities. To find these intersection points, we construct a new...

We construct examples of families of curves of genus 2 or 3 over Q whose Jacobians split completely and have various large rational torsion subgroups. For example, the rational points on a certain elliptic surface over P of positive rank parameterize a family of genus-2 curves over Q whose Jacobians each have 128 rational torsion points. Also, we find the genus-3 curve 15625(X + Y 4 + Z)− 96914...

We construct examples of families of curves of genus 2 or 3 over Q whose Jacobians split completely and have various large rational torsion subgroups. For example, the rational points on a certain elliptic surface over P of positive rank parameterize a family of genus-2 curves over Q whose Jacobians each have 128 rational torsion points. Also, we find the genus-3 curve 15625(X4 + Y 4 + Z4)− 969...

We consider the classical Diophantine problem of writing positive integers n as the sum of two rational cubes, i.e. n = x3 + y3 for x, y ∈ Q. A conjecture attributed to Sylvester asserts that a rational prime p > 3 can be so expressed if p ≡ 4, 7, 8 (mod 9). The theory of mock Heegner points gives a method for exhibiting such a pair (x, y) in certain cases. In this article, we give an expositor...

In this work we bring together classical mechanism theory with recent works in the area of Computer Aided Geometric Design(CAGD) of rational motions as well as curve approximation techniques in CAGD to study the problem of mechanism motion approximation from a computational geometric viewpoint. We present a framework for approximating algebraic motions of spherical mechanisms with rational B-Sp...