Estimation of the Bezout number for piecewise algebraic curve. (English) Zbl 1084.65014

Wang, Renhong; Xu, Zhiqiang

Sci. China, Ser. A 46, No. 5, 710-717 (2003).

Summary: 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 the four-color conjecture is also presented. By a Morgan-Scott triangulation, we will show the instability of the Bézout numbers of piecewise algebraic curves. By using the combinatorial optimization method, an upper bound of the Bézout number defined as the maximum finite number of intersection points of two piecewise algebraic curves is presented.

MSC:

65D07	Numerical computation using splines
14P05	Real algebraic sets
65D18	Numerical aspects of computer graphics, image analysis, and computational geometry

Keywords:

piecewise algebraic curve; Bézout theorem; bivariate splines; four-color conjecture; Morgan-Scott triangulation; Bézout number