Ringed surfaces and canal surfaces are surfaces that contain a one-parameter family of circles. Ringed surfaces can be described by a radius function, a directrix curve and vector field along the directrix curve, which specifies the normals of the planes that contain the circles. In particular, the class of ringed surfaces includes canal surfaces, which can be obtained as the envelopes of a one...

Faltings' theorem states that curves of genus g > 2 have finitely many rational points. Using the ideas of Faltings, Mumford, Parshin and Raynaud, one obtains an upper bound on the upper bound on the number of rational points [Szp85], XI, §2, but this bound is too large to be used in any reasonable sense. In 1985, Coleman showed [Col85] that Chabauty's method, which works when the Mordell-Weil ...

the main goal of this paper is to propose a new practical method for hydrologic soil group estimation in surface runoff process based on geomorphology facieses and slop. the determination of cn based on this method can reduce the time and cost of project. in this research the upstream rivers of glinak watershed have been used as a case study to examine the efficiency of proposed method with a c...

We give a characterization of the Deligne-Lusztig curve associated to the Suzuki group Sz(q) based on the genus and the number of Fq-rational points of the curve. §0. Throughout this paper by a curve we mean a projective, geometrically irreducible, and non-singular algebraic curve defined over the finite field Fq with q elements. Let Nq(g) denote the maximum number of Fq-rational points that a ...

Conic section is one of the geometric elements most commonly used for shape expression and mechanical accessory cartography. A rational quadratic Bézier curve is just a conic section. It cannot represent an elliptic segment whose center angle is not less than . However, conics represented in rational quartic format when compared to rational quadratic format, enjoy better properties such as bein...

The j-function acts as a parametrization of the classical modular curve. Its values at complex multiplication (CM) points are called singular moduli and are algebraic integers. A Shimura curve is a generalization of the modular curve and, if the Shimura curve has genus 0, a rational parameterizing function exists and when evaluated at a CM point is again algebraic over Q. This paper shows that ...

We combine various well-known techniques from the theory of heights, the theory of “noncritical Belyi maps”, and classical analytic number theory to conclude that the “ABC Conjecture”, or, equivalently, the so-called “Effective Mordell Conjecture”, holds for arbitrary rational points of the projective line minus three points if and only if it holds for rational points which are in “sufficiently...

We study the extension generated by the x-coordinates of the petorsion points of an elliptic curve over a function field of characteristic p. If S → C is a non-isotrivial elliptic surface in characteristic p with a pe-torsion section, then for pe > 11 our results imply restrictions on the genus, the gonality, and the p-rank of the base curve C, whereas for pe ≤ 11 such a surface can be construc...

An algebro-geometric method for determining the rational solvability of autonomous algebraic ordinary differential equations is extended from single equations of order 1 to systems of equations of arbitrary order but dimension 1. We provide necessary conditions, for the existence of rational solutions, on the degree and on the structure at infinity of the associated algebraic curve. Furthermore...

In this paper, we study fundamental properties of real curves, especially of rational real curves, and we derive several algorithms to decide the reality and rationality of curves in the complex plane. Furthermore, if the curve is real and rational, we determine a real parametrization. More precisely, we present a reality test algorithm for plane curves, and three different types of real parame...