Related papers: Inflection points and double tangents on anti-conv…

A positive integer $A$ is called a congruent number if $A$ is the area of a right-angled triangle with three rational sides. Equivalently, $A$ is a congruent number if and only if the congruent number curve $y^2=x^3-A^2x$ has a rational…

Given a real algebraic curve in the projective 3-space, its hyperbolicity locus is the set of lines with respect to which the curve is hyperbolic. We give an example of a smooth irreducible curve whose hyperbolicity locus is disconnected…

We give an explicit construction of a closed curve with constant torsion and everywhere positive curvature. We also discuss the restrictions on closed curves of constant torsion when they are constrained to lie on convex surfaces.

Let $Y$ be the complement of a plane quartic curve $D$ defined over a number field. Our main theorem confirms the Lang-Vojta conjecture for $Y$ when $D$ is a generic smooth quartic curve, by showing that its integral points are confined in…

Let $M$ be a compact closed manifold of variable negative curvature. Fix an element $\operatorname{id} \neq \gamma$ in the fundamental group $\Gamma$ of $M$, and denote the set of elements in $\Gamma$ that are conjugate to $\gamma$ by…

Given a submanifold $M\subset \mathbf{R}^\nu$, a curve $\gamma:I\to M$ and tangent vectors $v$ along $\gamma$, we roll the tangent space along $\gamma$. In doing so, we get an imprint/trace of $\gamma$ on the tangent space, as well as an…

Given an integer $\gamma\geq 2$ and an odd prime power $q$ we show that for every large genus $g$ there exists a non-singular curve $C$ defined over $\mathbb{F}_q$ of genus $g$ and gonality $\gamma$ and with exactly $\gamma(q+1)$…

We establish a congruence formula between $p$-adic logarithms of Heegner points for two elliptic curves with the same mod $p$ Galois representation. As a first application, we use the congruence formula when $p=2$ to explicitly construct…

Let $\Omega \subset R^n$, $n \geq 3$, be a fixed smooth bounded domain, and let $\gamma$ be a smooth conductivity in $\overline{\Omega}$. Consider a non-zero frequency $\lambda_0$ which does not belong to the Dirichlet spectrum of $L_\gamma…

We describe convex hulls of the simplest compact space curves, reducible quartics consisting of two circles. When the circles do not meet in complex projective space, their algebraic boundary contains an irrational ruled surface of degree…

It is proved that every convex body in the plane has a point such that the union of the body and its image under reflection in the point is convex. If the body is not centrally symmetric, then it has, in fact, three affinely independent…

The blown up complex projective plane in the twelve triple points of the dual Hesse arrangement has an infinite number of irreducible rational curves of self-intersection $-1$, for short, $(-1)$-curves. In the preprint version of [Dumnicki,…

We study real bitangents of real algebraic plane curves from two perspectives. We first show that there exists a signed count of such bitangents that only depends on the real topological type of the curve. From this follows that a generic…

We solve the following geometric problem, which arises in several three-dimensional applications in computational geometry: For which arrangements of two lines and two spheres in R^3 are there infinitely many lines simultaneously…

An n-gon is defined as a sequence \P=(V_0,...,V_{n-1}) of n points on the plane. An n-gon \P is said to be convex if the boundary of the convex hull of the set {V_0,...,V_{n-1}} of the vertices of \P coincides with the union of the edges…

Iommi & Kiwi showed that the Lyapunov spectrum of an expanding map need not be concave, and posed various problems concerning the possible number of inflection points. In this paper we answer a conjecture of Iommi & Kiwi by proving that the…

We prove an incidence theorem for points and planes in the projective space $\mathbb P^3$ over any field $\mathbb F$, whose characteristic $p\neq 2.$ An incidence is viewed as an intersection along a line of a pair of two-planes from two…

We investigate the 2-center problem for arbitrary strictly convex, centrally symmetric curves instead of usual circles. In other words, we extend the 2-center problem (from the Euclidean plane) to strictly convex normed planes, since any…

A smooth curve $\gamma: [0,1] \to S^2$ is locally convex if its geodesic curvature is positive at every point. J. A. Little showed that the space of all locally positive curves $\gamma$ with $\gamma(0) = \gamma(1) = e_1$ and $\gamma'(0) =…

Consider the locus of smooth curves of genus g and p-rank f defined over an algebraically closed field k of characteristic p. It is an open problem to classify which group schemes occur as the p-torsion of the Jacobians of these curves for…