On the number of Galois points for a plane curve in positive characteristic,II

For a smooth plane curve , we call a point a Galois point if the point projection at P is a Galois covering. We study Galois points in positive characteristic. We give a complete classification of the Galois group given by a Galois point and estimate the number of Galois points for C in most cases.      

Galois points for a plane curve in arbitrary characteristic

In 1996, Hisao Yoshihara introduced a new notion in algebraic geometry: a Galois point for a plane curve is a point from which the projection induces a Galois extension of function fields. Yoshihara has established various new approaches to algebraic geometry by using Galois point or generalized notions of it. It is an interesting problem to determine the distribution of Galois points for a given plane curve. In this paper, we survey recent results related to this problem.      

Rational points and Galois points for a plane curve over a finite field

We study the relationship between rational points and Galois points for a plane curve over a finite field. It is known that the set of Galois points coincides with that of rational points of the projective plane if the curve is the Hermitian, Klein quartic or Ballico–Hefez curve. The author proposes a problem: Does the converse hold true? If the curve of genus zero or one has a rational point, we have an affirmative answer.     

A birational embedding with two Galois points for quotient curves

A criterion for the existence of a birational embedding with two Galois points for quotient curves is presented. We apply our criterion to several curves, for example, some cyclic subcovers of the Giulietti–Korchmáros curve or of the curves constructed by Skabelund. New examples of plane curves with two Galois points are described, as plane models of such quotient curves.     

Galois points for double-Frobenius nonclassical curves

We determine the distribution of Galois points for plane curves over a finite field of q elements, which are Frobenius nonclassical for different powers of q. This family is an important class of plane curves with many remarkable properties. It contains the Dickson–Guralnick–Zieve curve, which has been recently studied by Giulietti, Korchmáros, and Timpanella from several points of view. A problem posed by the second author in the theory of Galois points is modified.     

On the Fundamental Groups of Galois Covering Spaces of the Projective Plane

We compute fundamental groups of the complements of a class of real curves in the complex projective plane. As a result, we obtain a new Zariski pair for arrangements of conics. As an application, we give a method for the computations of the fundamental groups of resolutions of Galois covering spaces of the projective plane ramifying along a special type of curves.     

一类Hammerstein积分方程的正分歧点

本文考虑有序Banach空间中形如x(t)=∫_Gk(t,s)[λAx(s) f(s,x(s),λ)]ds的Hammerstein型积分方程,给出了此类方程存在正分歧点的某些充分条件。     

Limits of Puiseux Expansions and Parametric Deformations of Plane Curve Singularities

We consider a family of parametrizations of unibranched plane curve singularities. Each member of the family can be expressed via the Puiseux expansion. Studying the limits of suitably renormalized coefficients of the Puiseux expansions, we obtain in some cases obstructions for a possible adjacency of unibranched plane curve singularities.

Furthermore, we show that the coefficients of the Puiseux expansion are not generic (in a precisely defined sense) as functions of the coefficients of parametrization of the singularity.     

On a Logarithmic Type Nonlocal Plane Curve Flow

In this paper the author devotes to studying a logarithmic type nonlocal plane curve flow.Along this flow,the convexity of evolving curve is preserved,the perimeter decreases,while the enclosed area expands.The flow is proved to exist globally and converge to a finite circle in the C∞metric as time goes to infinity.     

空间平面内整点n边形上的整点个数

本文给出了计算空间平面内整点n边形上的整点个数的一个公式。     

1－集压缩映射的非零不动点

本文给出了１－集压缩映射的一些新的非零不动点定理，它推广和改进了［１，２，４，５］中的某些重要定理．     

平面内判断点与封闭区域关系的快速算法

在诸多研究领域中,判断点是否在封闭多边形区域内是一个非常基本而重要的问题.在分析解决这一问题的常用方法的基础上,以矢量叉积结果的正负号为研究对象,分类讨论特殊情形,提出完整判断点与封闭多边形(包括曲线边)区域关系的算法,然后用LabVIEW编程对算法进行验证并对算法所执行的时间进行测试分析.测试结果显示多边形的凸点数及...     

On the Curve X(9)

Computations using theta functions with characteristic show that the modular curve X(9) is the complete intersection of two cubics. The holomorphic differentials and Weierstrass gap sequence are also computed.     

Positive Solutions and Extremal Points for Differential Equations

Extremal point properties are examined for the boundary value problem x⁽ⁿ⁾ + ? ^n-2 _i=0 A_i(t)x⁽ⁱ⁾ = 0, x⁽ⁱ⁾(0) = x^(n-2)(T) = 0, 0?i?n-2, where the A_i's, a characterization for the first extremal point is given in terms of the existence of a solution which is positive with respect to a cone in a Banach space. Also, an existence theorem is obtained for a related nonlinear problem     

关于平面曲线的分形维数

关于平面曲线的分形维数邓冠铁（武汉华中理工大学教学系，武汉４３００７４）关键词分形维数，平面曲线，周期函数．分类号ＡＭＳ（１９９１）２８Ａ８０／ＣＣＬＯ１７４．１２为方便起见，采用如下定义和符号：设为平面上有界非空集，对ε＞０，用Ｇ（ε）＝｛（ｘ，ｙ...     

椭圆曲线y~2=x~3-21x+90的正整数点

利用初等方法证明了椭圆曲线y~2=x~3-21x+90无正整数点.     

On the combinatorics of Galois numbers

We define interval decompositions of the lattice of subspaces of a finite-dimensional vector space. We show that such a decomposition exists if and only if there exists a family of linear forms with certain properties. As applications we prove that all finite-dimensional real vector spaces admit an interval decomposition, while GF(2)ⁿ has an interval decomposition if and only if n≤4. On the other hand, we present an interval decomposition of GF(3)⁵. This partially answers a question of Faigle and Kruse (2004) [1] and [4].