折叠点下曲线的相交重数

The intersection multiplicity of curves under the fold point

解析几何中最重要的基本问题之一是求两条代数曲线的交点数。Bézout定理表明,m次代数曲线和n次代数曲线有mn个交点计算重数,除非它们有共同的分量,否则交点个数不超过mn。在局部情况下,Liang分别在R^(2)和P^(2)_(R)中介绍了两条代数曲线在一点的相交重数。由于相交重数与折叠点有密切联系,而线性变换(或射影变换)又不改变R^(2)(或P^(2)_(R))中曲线的相交重数,所以探讨变换后折叠点重数的变化规律具有一定的研究意义。文章分别研究了R^(2)和P^(2)_(R)中曲线的相交重数,利用R^(2)(或P^(2)_(R))中的线性变换(或射影变换)给出曲线在一点相交重数变换关系的等价性。

In analytic geometry,one of the most important fundamental problems is to find the number of intersection points of two algebraic curves.Bézout s theorem states that two algebraic curves of degrees m and n intersect at mn points,counting multiplicities,and cannot meet at more than mn points unless they have a component in common.In the local case,Liang introduced the intersection multiplicity of two algebraic curves at some point in R^(2) and P^(2)_(R) respectively.Since the intersection multiplicity is closely related to the fold point,and linear transformation(or projective transformation)preserves the intersection multiplicity of curves in R^(2)(resp.P^(2)_(R)),it is of certain research significance to discuss the change rule of the multiplicity of the fold point after transformation.In this paper,we study the intersection multiplicity of curves at a point in R^(2) and P^(2)_(R),respectively.We give the equivalence of transformation relation of the intersection multiplicity of curves at a point by linear transformation(resp.projective transformation)in R^(2)(resp.P^(2)_(R)).