Solving parametric piecewise polynomial systems

• Authors: Yisheng Lai;Renhong Wang;Jinming Wu
• Venue: Journal of Computational and Applied Mathematics
• Year: 2011

We deal with C^r smooth continuity conditions for piecewise polynomial functions on @D, where @D is an algebraic hypersurface partition of a domain @W in R^n. Piecewise polynomial functions of degree, at most, k on @D that are continuously differentiable of order r form a spline space C"k^r. We present a method for solving parametric systems of piecewise polynomial equations of the form Z(f"1,...,f"n)={X@?@W|f"1(V,X)=0,...,f"n(V,X)=0}, where f"@w@?C"k"""@w^r^"^@w(@D), and f"@w|"@s"""i@?Q[V][X] for each n-cell @s"i in @D, V=(u"1,u"2,...,u"@t) is the set of parameters and X=(x"1,x"2,...,x"n) is the set of variables; @s"1,@s"2,...,@s"m are all the n-dimensional cells in @D and @W=@?"i"="1^m@s"i. Based on the discriminant variety method presented by Lazard and Rouillier, we show that solving a parametric piecewise polynomial system Z(f"1,...,f"n) 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(f"1,...,f"n). 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 @D.