Solving parametric piecewise polynomial systems

摘要

We deal with Cr smooth continuity conditions for piecewise polynomial functions on Δ, where Δ is an algebraic hypersurface partition of a domain Ω in Rn. Piecewise polynomial functions of degree, at most, k on Δ that are continuously differentiable of order r 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] for each n-cell σi in Δ, V=(u1,u2,…,uτ) is the set of parameters and X=(x1,x2,…,xn) is the set of variables; σ1,σ2,…,σm are all the n-dimensional cells in Δ 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) is reduced to the computation of discriminant variety of Z. 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). 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 n-cells in Δ.