An Upper Bound of the Bezout Number for Piecewise Algebraic Curves over a Rectangular Partition

A piecewise algebraic curve is a curve defined by the zero set of a bivariate spline function. Given two bivariate spline spaces (Δ) and (Δ) over a domain D with a partition Δ, the Bezout number BN(m,r;n,t;Δ) is defined as the maximum finite number of the common intersection points of two arbitrary piecewise algebraic curves and , where ∈ (Δ) and ∈ (Δ). In this paper, an upper bound of the Bezout number for piecewise algebraic curves over a rectangular partition is obtained. 1. Introduction Let be a bounded domain, and let be the collection of real bivariate polynomials with total degree not greater than . Divide by using finite number of irreducible algebraic curves we get a partition denoted by . The subdomains are called the cells. The line segments that form the boundary of each cell are called the edges. Intersection points of the edges are called the vertices. The vertices in the inner of the domain are called interior vertices, otherwise are called boundary vertices. For a vertex , its so-called star means the union of all cells in sharing as a common vertex, and its degree is defined as the number of the edges sharing as a common endpoint. If is odd, we call an odd vertex. For integers and with , the bivariate spline space with degree and smoothness over with respect to is defined as follows [1, 2]: For a spline , the zero set is called a piecewise algebraic curve [1, 2]. Obviously, the piecewise algebraic curve is a generalization of the usual algebraic curve [3, 4]. In fact, the definition of piecewise algebraic curve is originally introduced by Wang in the study of bivariate spline interpolation. He pointed out that the given interpolation knots are properly posed if and only if they are not lying on a same nonzero piecewise algebraic curve [1, 2]. Hence, to solve a bivariate spline interpolation problem, it is necessary to deal with the properties of piecewise algebraic curve. Moreover, piecewise algebraic curve is also helpful for us to study the usual algebraic curve. Besides, piecewise algebraic curve also relates to the remarkable Four-Color conjecture [5–7]. In fact, the Four-Color conjecture holds if and only if, for any triangulation, there exist three linear piecewise algebraic curves such that the union of them equals the union of all central lines of all triangles in the triangulation. We know that any triangulation is 2-vertex signed [6, 7], which means the vertices of the triangulation can be marked by or such that the vertices of every triangle in the triangulation are marked by different numbers. So, we remark that the