Solving parametric piecewise polynomial systems

We deal with CrCr smooth continuity conditions for piecewise polynomial functions on ΔΔ, where ΔΔ is an algebraic hypersurface partition of a domain ΩΩ in RnRn. Piecewise polynomial functions of degree, at most, kk on ΔΔ that are continuously differentiable of order rr form a spline space Ckr.We present a method for solving parametric systems of piecewise polynomial equations of the form Z(f1,…,fn)={X∈Ω∣f1(V,X)=0,…,fn(V,X)=0}, where fω∈Ckωrω(Δ), and fω∣σi∈Q[V][X]fω∣σi∈Q[V][X] for each nn-cell σiσi in ΔΔ, V=(u1,u2,…,uτ)V=(u1,u2,…,uτ) is the set of parameters and X=(x1,x2,…,xn)X=(x1,x2,…,xn) is the set of variables; σ1,σ2,…,σmσ1,σ2,…,σm are all the nn-dimensional cells in ΔΔ and Ω=⋃i=1mσi.Based on the discriminant variety method presented by Lazard and Rouillier, we show that solving a parametric piecewise polynomial system Z(f1,…,fn)Z(f1,…,fn) is reduced to the computation of discriminant variety of ZZ. The variety can then be used to solve the parametric piecewise polynomial system.We also propose a general method to classify the parameters of Z(f1,…,fn)Z(f1,…,fn). This method allows us to say that if there exist an open set of the parameters’ space where the system admits exactly a given number of distinct torsion-free real zeros in every nn-cells in ΔΔ.