Research On Bezout's Number Of Piecewise Algebraic Curves

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. It is obvious that the piecewise algebraic curve is a generalization of the classical algebraic curve. The piecewise algebraic curve is not only very important for the interpolation by the bivariate splines (cf. [20]), but also a useful tool for studying traditional algebraic curves. It is well known that Bezout' s theorem is an important and classical theorem in the algebraic geometry (cf. [19]). Its weak form says that two algebraic curves will have infinitely many intersection points provided that the number of their intersection points more than the product of their degrees. Denote by BN(m, 0; n, 0; △) the so-called Bezout' s number. It means any two piecewise algebraic curves f(x,y) = 0,g(x,y) = 0, f ∈Srm(△),g∈Sln(△), must have infinitely many intersection points providedthat they have more than BN intersection points.In cite [22], the upper boundary of BN(m,0;n,0;△)is presented. In [21], an upper boundary of BN(m,1;n,1;△) is presented. In [9] Xu used the combinatorial method which is different with the method in [22], an upper bound of the BN(m,r;n,t;△) is presented.In this thesis, we make some improvement to BN(m,r;n,t;△), which is obtained by [9]. And with the further studying of BN(m,0;n,0;△), obtained the sufficient and necessary condition ofBN(m,r;n,t;△) = mnT, And using this character illustrated one interesting factor of graphics .Newton Formula is an important formula of combine mathematics, In this thesis we obtained one expression of sn+1 = x1n+1+ x2n+1 + …… + xnn+1 by using NewtonFormula, And using this expression, we estimated the upper boundary of zero points of homogeneous triangular spline.