分片代数曲线Bezout数的估计

分片代数曲线定义为二元样条函数的零点集合.首先证明了关于三角剖分的一个猜想. 随后,指出了分片线性代数曲线与四色猜想之间的内在联系.通过经典的Morgan-Scott剖分,指出分片代数曲线的ezout数的不稳定性.利用组合优化方法,得到任意阶光滑分片代数曲线的Bezout数的上界.这个上界不仅适用于三角剖分,而且对任意网线为直线段的剖分均成立.     

拟贯穿剖分上分片代数曲线的Nother型定理

代数曲线的Nother定理是代数几何中经典并且十分重要的结论.作为二元样条的零点集,分片代数曲线是经典代数曲线的推广.分片代数曲线的Nother型定理对研究二元样条空间的Lagrange插值有至关重要的作用.利用拟贯穿剖分的特点、二元样条的性质与代数几何的相关知识,给出了拟贯穿剖分上分片代数曲线的Nother型定理.     

三角剖分上分片代数曲线的Nöther型定理

分片代数曲线是经典代数曲线的推广. 贯穿剖分上的分片代数曲线的Nöther型定理对构造二元样条空间的Lagrange插值适定结点组有非常重要的作用. 文中利用二元样条的性质, 给出了任意三角剖分上分片代数曲线的Nöther型定理.     

实分片代数曲线的拓扑结构

The piecewise algebraic curve is a kind generalization of the classical algebraic curve.By analyzing the topology of real algebraic curves on the triangles,a practi-caUy algrithm for analyzing the topology of piecewise algebraic curves is given.The algrithm produces a planar graph which is topologically equivalent to the piecewise algebraic curve.     

三角剖分上分片代数曲线的Nther型定理

分片代数曲线是经典代数曲线的推广．贯穿剖分上的分片代数曲线的Nther型定理对构造二元样条空间的Lagrange插值适定结点组有非常重要的作用．文中利用二元样条的性质,给出了任意三角剖分上分片代数曲线的N(?)ther型定理．     

分片代数集合的不可约性和同构

本文利用多元样条函数来定义分片代数集合,讨论了分片代数集合的不可约性和同构问题,给出了分片代数集合不可约的两个等价条件,并把分片代数集合的同构分类问题转化为交换代数的同构分类问题。     

三角剖分上分片代数曲线的N(o)ther型定理

分片代数曲线是经典代数曲线的推广.贯穿剖分上的分片代数曲线的N(o)ther型定理对构造二元样条空间的Lagrange插值适定结点组有非常重要的作用.文中利用二元样条的性质,给出了任意三角剖分上分片代数曲线的N(o)ther型定理.     

代数曲线渐近线的一种求法

一、渐近线的极限定义在拓广平面内的解释在数学分析中,渐近线的定义为: 如果曲线上一点沿曲线趋于无穷远时,该点与某直线的距离趋于0,则此直线称为曲线的渐近线。在这个定义中, (1) 曲线上的点趋于无穷远可以分为x→     

实分片代数曲线的某些研究

分片代数曲线作为二元样条函数的零点集合是经典代数曲线的推广. 利用代数的基本知识, 本文对实分片代数曲线的基本性质进行了初步讨论, 并且将实分片代数曲线与相应的二元样条分类进行讨论. 最后, 对实分片代数曲线上的孤立点进行了研究.     

The Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, we propose the Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations. We show that, if two continuous piecewise algebraic curves of degrees m and n respectively meet at mnT distinct points over a cross-cut triangulation, where T denotes the number of cells of the triangulation, then any continuous piecewise algebraic curve of degree m + n − 2 containing all but one point of them also contains the last point.     

Foci and Foliations of Real Algebraic Curves

We discuss diverse results whose common thread is the notion of focus of an algebraic curve. In a unified setting, which combines elements of projective geometry, complex analysis and Riemann surface theory, we explain the roles of ordinary and singular foci in results on numerical ranges of matrices, quadrature domains, Schwarzian reflection, and other topics. We introduce the notion of canonical foliation of a real algebraic curve, which places foci into the context of continuous families of plane curves and provides a useful method of visualization of all relevant structures in a planar graphical image. Lecture held by Joel Langer in the Seminario Matematico e Fisico on October 6, 2006 Received: July 2007     

The Automorphism Group of Plane Algebraic Curves with Singer Automorphisms

The main result in Cossidente and Siciliano (J. Number Theory, Vol. 99 (2003) pp. 373–382) states that if a Singer subgroup of PGL(3,q) is an automorphism group of a projective, geometric irreducible, non-singular plane algebraic curve then either or . In the former case is projectively equivalent to the curve with equation X^q+1Y+Y^q+1+X=0 studied by Pellikaan. Furthermore, the curve has a very nice property from Finite Geometry point of view: apart from the three distinguished points fixed by the Singer subgroup, the set of its -rational points can be partitioned into finite projective planes . In this paper, the full automorphism group of such curves is determined. It turns out that is the normalizer of a Singer group in .     

Modulus of Continuity of Piecewise Analytic Functions

We obtain conditions under which the modulus of continuity of a piecewise analytic function given on a closed interval of the real axis is an analytic function in a neighborhood of zero.     

The Bezout Number for Piecewise Algebraic Curves

The computation of the Bezout number, the maximum number of intersection points between two piecewise algebraic curves whose common points are finite, is considered. A piecewise algebraic curve is a curve determined by a bivariate spline function. It is found that the maximum number of intersections depends not only on the degrees and the differentiability of the spline functions, but also on the structure of the partition on which the spline functions are defined.     

Height Uniformity for Algebraic Points on Curves

We recall the main result of L. Caporaso, J. Harris, and B. Mazur's 1997 paper of Uniformity of rational points. It says that the Lang conjecture on the distribution of rational points on varieties of general type implies the uniformity for the numbers of rational points on curves of genus at least 2. In this paper we will investigate its analogue for their heights under the assumption of the Vojta conjecture. Basically, we will show that the Vojta conjecture gives a naturally expected simple uniformity for their heights.     

The Bezout number of piecewise algebraic curves

Based on the discussion of the number of roots of univariate spline and the common zero points of two piecewise algebraic curves, a lower upbound of Bezout number of two piecewise algebraic curves on any given non-obtuse-angled triangulation is found. Bezout number of two piecewise algebraic curves on two different partitions is also discussed in this paper.     

Integrable Deformations of Algebraic Curves

We present a general scheme for determining and studying integrable deformations of algebraic curves, based on the use of Lenard relations. We emphasize the use of several types of dynamical variables: branches, power sums, and potentials.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 94–101, July, 2005.