The Variation Diminishing Property Of Triangular Splines

A piecewise algebraic curve is a curve determined by the zero set of a bivariate splinefunction. It is obvious that the piecewise algebraic curve is a generalization of the classicalalgebraic curve. The piecewise algebraic curve is not only very important for the interpolationby the bivariate splines (cf. [34]), but also a useful tool for studying traditional algebraiccurves. It is well known that Bezout theorem is an important and classical theorem in thealgebraic geometry (cf. [37]). Its weak form says that two algebraic curves will have infinitelymany intersection points provided that the number of their intersection points is more than theproduct of their degree. Denote by BN= BN(m,r;n,t;A) the so-called Bezout number. Itmeans that any two piecewise algebraic curves f(x,y)=0,g(x,y)=0,f∈S_m~r(Δ),g∈S_n~t (Δ)must have infinitely many intersection points provided that they have more than BNintersection points.In cite [22], the upper bound of BN(m,0;n,0;Δ) is presented. In cite [23], the upperbound of BN(m,1;n,1;A) is presented. In cite [24], Xu using the combinatorial methodwhich is different from the method in [22], gave an upper bound of the BN(m, r; n, t;Δ).The above introduction is about the Bezout number of the piecewise algebraic curves. Infact, the Bezout number is not the only focus in the piecewise algebraic curves. Nothertheorem, Riemann-Roch theorem, Cayley-Bacharach theorem and so on also play importantpart in the algebraic curves and piecewise algebraic curves. Lai([30]) and Zhu([32])separately extended the Nother theorem, Riemann-Roch theorem and Cayley-Bacharachtheorem from the algebraic curves into the piecewise algebraic curves. These can bring thetheoretical and applied values in the development of the piecewise algebraic curves.In this thesis, we have proved the Variation Diminishing Property of Triangular Splines.According to this, we can get an upper bound of the BN(m, r; n,t;Δ) in the star region.