Related papers: A Characterization of Quintic Helices
The aim of this note is to give a surprising symmetry property of some harmonic algebraic curves: when all the roots $z_i$ of a complex polynomial $P$ lie on the unit circle $\U$, the points of $\U$ different from the $z_i$, and such that…
A graph is called a nut graph if zero is its eigenvalue of multiplicity one and its corresponding eigenvector has no zero entries. A graph is a bicirculant if it admits an automorphism with two equally sized vertex orbits. There are four…
We complete the topological classification of real algebraic non-singular curves of bidegree $(5, 5)$ on the quadric ellipsoid. We show in particular that previously known restrictions form a complete system for this bidegree. Therefore,…
A general scheme for determining and studying integrable deformations of algebraic curves is presented. The method is illustrated with the analysis of the hyperelliptic case. An associated multi-Hamiltonian hierarchy of systems of…
The action of ring automorphisms of the polynomial ring in two variables over the real numbers on real plane curves is considered. The orbits containing degree-three polynomials are computed, with one representative per orbit being…
We give two formulae which express the Alexander polynomial $\Delta^C$ of several variables of a plane curve singularity $C$ in terms of the ring ${\cal O}_{C}$ of germs of analytic functions on the curve. One of them expresses $\Delta^C$…
A plane curve $C$ in $\mathbb{P}^2$ defined over $\mathbb{F}_q$ is called plane-filling if $C$ contains every $\mathbb{F}_q$-point of $\mathbb{P}^2$. Homma and Kim, building on the work of Tallini, proved that the minimum degree of a smooth…
We consider the problem of classifying those graphs that arise as an undirected square of an oriented graph by generalising the notion of quasi-transitive directed graphs to mixed graphs. We fully classify those graphs of maximum degree…
A non-singular connected algebraic curve $A$ in a simply connected algebraic surface $X$ can be knotted so that its homology class and the fundamental group of its complement in $X$ is preserved, provided $A$ is sufficiently complex (not…
Exact solution to many problems in mathematical physics and quantum field theory often can be expressed in terms of an algebraic curve equipped with a meromorphic differential. Typically, the geometry of the curve can be seen most clearly…
The paper is a contribution of the conjecture of Kobayashi that the complement of a generic plain curve of degree at least five is hyperbolic. The main result is that the complement of a generic configuration of three quadrics is hyperbolic…
In this article we introduce Hecke operators on the differential algebra of geometric quasi-modular forms. As an application for each natural number $d$ we construct a vector field in six dimensions which determines uniquely the polynomial…
In this paper, we contribute toward a classification of two-variable polynomials by classifying (up to an automorphism of $C^2$) polynomials whose Newton polygon is either a triangle or a line segment. Our classification has several…
In this paper, we call a sub-scheme of dimension one in $\mathbb{P}^3$ a curve. It is well known that the arithmetic genus and the degree of an aCM curve $D$ in $\mathbb{P}^3$ is computed by the $h$-vector of $D$. However, for a given curve…
Let K be a field of characteristic different from 2 and C an elliptic curve over K given by a Weierstrass equation. To divide an element of the group C by 2, one must solve a certain quartic equation. We characterise the quartics arising…
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…
We show that the degree of the Alexander polynomial of an irreducible plane algebraic curve with nodes and cusps as the only singularities does not exceed ${5 \over 3}d-2$ where $d$ is the degree of the curve. We also show that the…
When the number of non-triangular faces adjacent to a vertex $v$ is less than or equal to three, the vertex $v$ will be called (\emph{combinatorially}) \emph{rigid}. We study the number of rigid vertices and suggest a conjecture on a…
In n-dimensional Euclidean space E^n, harmonic curvatures of a non-degenerate curve defined by \"Ozdamar and Hacisaliho\u{g}lu [4]. In this paper, We define a new type of curves called LC helix when the angle between tangent of this curve…
A Chebyshev curve C(a,b,c,\phi) has a parametrization of the form x(t)=Ta(t); y(t)=T_b(t) ; z(t)= Tc(t + \phi), where a,b,c are integers, Tn(t) is the Chebyshev polynomial of degree n and \phi \in \RR. When C(a,b,c,\phi) has no double…