The Estimation Of Bezout Numbers For The Piecewise Algebraic Curves

The piecewise algebraic curve is defined as the set of zero of a binary spline function.It's a natural extension of classical algebraic curves.Due to the difficulty of the piecewise algebraic curve,We know a little about its properties.Bezout theorem is the open volume theorem of traditional algebraic geometry.The spread of the Bezout theorem?the number of finite intersection points with the largest number of two-segment algebra curves?is signify for the study of piecewise algebraic curve.Many applications of the classic algebraic curve theory require the solution of Bezout's theorem.Any substantial development of the Bezout number will have a profound effect on the PAC.Through the knot theorem,using MATLAB and the Descartes symbol rule to derive the Bezout number of low-order piecewise algebraic curves on ordinary.However,when the knot theorem transforms two binary spline with different smoothness into a new unary spline function,the smoothness of the unary spline function is determined by the original low smoothness.Therefore,the knot theorem cannot reflect the smoothness of the original two splines.Below we try the new two methods.The first method:geometry-based combinatorial optimization,the number of roots of the unary spline is obtained by the Descartes symbol rule.On this basis,using the homogeneous triangle polynomial conclusion to get upper bound of the binary spline,the upper bound of Bezout of under parallel subdivision is obtained by using combination optimization and lingo software.The method is proved to be effective in practice.The second method:graph theory.By using the necessary and sufficient conditions for a series of straight line segments on arbitrary triangulation to form theS₁⁰-piecewise algebraic curve to prove our-color conjecture is equivalent to a factorization conjecture of a planar graph without bridge.And some new properties of triangulation are given.It provides a bridge between graph theory and piecewise algebraic curves,It provides more possibilities for Bezout Numbers of Piecewise algebraic curve.Figure 12;Table 0;Reference 52.