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 practically 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.

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Bezout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。

In this paper, some important properties of orthogonal polynomials of two variables are investigated. The concepts of invariant factor for orthogonal polynomials of two variables are introduced. The presented results include Stieltjies type theorems for multivariate orthogonal polynomials and the corresponding asymptotic expansion formulas.

It is well known that splines play an important role in many 0elds, especially, their close relationship with wavelets makes them have more widespread applications in numerous scienti0c and engineering domains. Univariate and bivariate splines have been well studied and lots of results have been obtained. Because of the intrinsic diffculty between bivariate case and higher-dimension (three or more dimensions) settings, the study of splines on higher dimensions are very limited. For example, the study of the bivariate splines on a three-direction mesh triangulation has obtained many important and excellent results, but almost all of those results have no analog generalization to higher dimensions. In this paper, we will study the higher-dimension splines de0ned on n+1 mesh simplical partitions which is the analog of bivariate splines on three-mesh triangulations. We have also pointed out many interesting di6erences between bivariate splines and higher-dimensional cases. Our main results are that, similar to bivariate and trivariate cases, a necessary and suffcient condition for Sr k (△) to contain a B-spline is k≥1 2(r+1) (n+1) for r≥1 being odd and k≥1 2r(n+1)+1 for r≥0 being even.

A piecewise algebraic curve is de1ned by a bivariate spline function.Using the techniques of the B-net form of bivariate splines function, discriminant sequence of polynomial (cf.Yang Lu et al.(Sci.China Ser.E 39(6) (1996) 628) and Yang Lu et al.(Nonlinear Algebraic Equation System and Automated Theorem Proving, Shanghai Scienti1c and Technological Education Publishing House, Shanghai, 1996)) and the number of sign changes in the sequence of coeffecients of the highest degree terms of sturm sequence, we determine the number of real intersection points of two piecewise algebraic curves whose common points are 1nite.A lower bound of the number of real intersection points is given in terms of the method of rotation degree of vector field.