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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.