Estimation of the Bezout number for piecewise algebraic curve

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, a conjecture on triangulation is confirmed. The relation between the piecewise linear algebraic curve and four-color conjecture is also presented. By Morgan-Scott triangulation, we will show the instability of Bezout number of piecewise algebraic curves. By using the combinatorial optimization method, an upper bound of the Bezout number defined as the maximum finite number of intersection points of two piecewise algebraic curves is presented.