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.

Dianxuan Gong, Born in February 1981. Studied Computational geometry at Dalian University of Technology and got the doctor's degree in 2009. Major in Spline function theory, Piecewise algebraic curves, Piecewise algebraic variety. Now work in College of Sciences, North China University of Science and Technology,Tangshan, China.