This paper establishes the basic theory on piecewise polynomial functions that are with different smoothness degree over the whole domain, and presents the theory and methods for solving zero-dimensional parametric piecewise polynomial systems. It shows that solving such a system amounts to solving parametric semi-algebraic systems, which is then reduced to the computation of discriminant varieties. Here, is the number of -dimensional cells in the hereditary partition of the domain. The parametric piecewise polynomial system is ultimately solved utilizing the critical points method and the Collins partial cylindrical algebraic decomposition method.
The paper also proposes a classification method and its algorithm to address whether there exists an open set in the parameter domain such that, for each point in the open set, the corresponding zero-dimensional non-parametric piecewise polynomial system (a realization of the parametric piecewise polynomial system) has exactly the given number of torsion-free real zeros in the cells respectively.