An upper bound of the Bezout number for piecewise algebraic curves

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. Based on the discussion of the number of the zeros of homogeneous trigonometric splines with different smoothness and the common points of two piecewise algebraic curves over a star partition, a better upper bound of Bezout number of two piecewise algebraic curves over arbitrary triangulation is found. Moreover, upper bounds of the Bezout number BN(m,r;n,r;Δ)BN(m,r;n,r;Δ) for piecewise algebraic curves over several special partitions such as rectangular partition, type-1 triangulation and type-2 triangulation are obtained.