Related papers: A Characterization of Quintic Helices

It is a classical fact going back to F. Klein that an elliptic curve $E$ over $\bar{\mathbb{Q}}$ is defined by a homogeneous polynomial in $3$ variables with coefficients in $\mathbb{Q}(j_{E})$, where $j_{E}$ is the $j$-invariant of $E$,…

Boix, De Stefani and Vanzo have characterized ordinary/supersingular elliptic curves over $\mathbb{F}_p$ in terms of the level of the defining cubic homogenous polynomial. We extend their study to arbitrary genus, in particular we prove…

This text is a presentation of a set of formulae, first found by Vainsencher (for $\delta \leq 6$) and shortly after improved by Kleiman and Piene, counting $\delta$-nodal curves in a complete linear system on a smooth surface, if $\delta…

We construct a polynomial planar vector field of degree two with one invariant algebraic curves of large degree. We exhibit an explicit quadratic vector fields which invariant curves of degree nine, twelve, fifteen and eighteen degree.

In this note, we consider M-curves of odd degree with real scheme of the form $< \mathcal{J} \amalg \alpha \amalg 1 < \beta > >$. With help of complex orientations, we prove that for $m = 9$, $\alpha \geq 2$, and for $m = 11$, $\alpha \geq…

Let K be a finite field. We know that a half of elements of K* is a square. So it is natural to ask how many of them appear as x-coordinate of points on an elliptic curve over K. We consider a specific class of elliptic curves over finite…

The degree of a polynomial representing (or approximating) a function f is a lower bound for the number of quantum queries needed to compute f. This observation has been a source of many lower bounds on quantum algorithms. It has been an…

Call a colouring of a graph distinguishing, if the only colour preserving automorphism is the identity. A conjecture of Tucker states that if every automorphism of a graph $G$ moves infinitely many vertices, then there is a distinguishing…

We study the finiteness of low degree points on certain modular curves and their Atkin--Lehner quotients, and, as an application, prove the modularity of elliptic curves over all but finitely many totally real fields of degree $5$. On the…

We determine which of the modular curves $X_\Delta(N)$, that is, curves lying between $X_0(N)$ and $X_1(N)$, are bielliptic. Somewhat surprisingly, we find that one of these curves has exceptional automorphisms. Finally we find all…

We study a family of polynomials in two variables having moduli up to bilipschitz equivalence: two distinct polynomials of this family are not bilipschitz equivalent. However any level curve of the first polynomial is bilipschitz equivalent…

Let $C$ be a non-hyperelliptic algebraic curve. It is known that its canonical image is the intersection of the quadrics that contain it, except when $C$ is trigonal (that is, it has a linear system of degree 3 and dimension 1) or…

A Chebyshev curve $\mathcal{C}(a,b,c,\phi)$ has a parametrization of the form$ x(t)=T\_a(t)$; \ $y(t)=T\_b(t)$; $z(t)= T\_c(t + \phi)$, where $a,b,c$are integers, $T\_n(t)$ is the Chebyshev polynomialof degree $n$ and $\phi \in \mathbb{R}$.…

The symmetric edge polytope of a simple graph is a lattice polytope defined as the convex hull of a subset of the type A roots corresponding to the edges of the graph. In this article we prove a sharp lower bound for the number of edges of…

We prove that every rational trinomial affine hypersurface admits a horizontal polynomial curve. This result provides an explicit non-trivial polynomial solution to a trinomial equation. Also we show that a trinomial affine hypersurface…

Let a set of nodes $\mathcal X$ in the plane be $n$-independent, i.e., each node has a fundamental polynomial of degree $n.$ Assume that $\#\mathcal X=d(n,k-3)+3= (n+1)+n+\cdots+(n-k+5)+3$ and $4 \le k\le n-1.$ In this paper we prove that…

Semiholomorphic polynomials are functions $f:\mathbb{C}^2\to\mathbb{C}$ that can be written as polynomials in complex variables $u$, $v$ and the complex conjugate $\overline{v}$. We prove the semiholomorphic analogoue of Akbulut's and…

In this paper, we first find an estimate for the range of polyharmonic mappings in the class $HC_{p}^{0}$. Then, we obtain two characterizations in terms of the convolution for polyharmonic mappings to be starlike of order $\alpha$, and…

We prove that a curve of degree $dk$ on a very general surface of degree $d \geq 5$ in $\mathbb{P}^3$ has geometric genus at least $\frac{dk(d-5)+k}{2} + 1$. This improves bounds given by G. Xu. As a corollary, we conclude that the very…

Every graph of size $q$ (the number of edges) and minimum degree $\delta$ is hamiltonian if $q\le\delta^2+\delta-1$. The result is sharp.