Some research on piecewise algebraic variety and its application

This book is a summary of the author''s research work in recent years. It is mainly about the theoretical and computational aspects of the piecewise algebraic variety. The main problems are: the Bezout number of piecewise algebraic curve the lower bound on the number of real intersection points of piecewise algebraic curves piecewise algebraic curve and four-color theorem generalization of BKK theorem in multivariate spline space and the Minkowski decomposition of convex polyhedron. Some basic knowledge is added to facilitate the reading.