Real zeros of the zero-dimensional parametric piecewise algebraic variety

The piecewise algebraic variety is the set of all common zeros of multivariate splines. We show that solving a parametric piecewise algebraic variety amounts to solve a finite number of parametric polynomial systems containing strict inequalities. With the regular decomposition of semi-algebraic systems and the partial cylindrical algebraic decomposition method, we give a method to compute the supremum of the number of torsion-free real zeros of a given zero-dimensional parametric piecewise algebraic variety, and to get distributions of the number of real zeros in every n-dimensional cell when the number reaches the supremum. This method also produces corresponding necessary and sufficient conditions for reaching the supremum and its distributions. We also present an algorithm to produce a necessary and sufficient condition for a given zero-dimensional parametric piecewise algebraic variety to have a given number of distinct torsion-free real zeros in every n-cell in the n-complex.