Progress in construction of piecewise algebraic hypersurfaces and solving parametric piecewise polynomial systems

The multivariate spline is a piecewise polynomial with certain smoothness, and the piecewise algebraic hypersurfaces with certain smoothness (i.e. the set of all common zeros of multivariate splines) have become useful tools for representing or approximating surfaces. Studying an effective method for the construction of real piecewise algebraic hypersurfaces of a given degree with certain smoothness and prescribed topology is one of the important ways to solve the problem of how to represent or approximate geometric objects with certain topological structure (especially the complex topology structure), and also a new and important topic of computational geometry and algebraic geometry. Parametric piecewise polynomial systems are not only closely related to a series of reasearch, such as intersection of surfaces; blending curves and surfaces; generation of transition surfaces, but also essential generalization of the parametric semi-algebraic systems. The purpose of this paper is to introduce some recent research progress on the construction of real piecewise algebraic hypersurfaces, and parametric piecewise polynomial systems.