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TitleSolving parametric piecewise polynomial systems
Author(s) Yi-Sheng Lai, Ren-Hong Wang, Jinzhao Wu
TypeArticle in Journal
AbstractWe deal with C r smooth continuity conditions for piecewise polynomial functions on Δ , where Δ is an algebraic hypersurface partition of a domain Ω in R n . Piecewise polynomial functions of degree, at most, k on Δ 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 ∈ Ω ∣ f 1 ( V , X ) = 0 , … , f n ( V , X ) = 0 , where f ω ∈ C k ω r ω ( Δ ) , and f ω ∣ σ i ∈ Q [ V ] [ X ] for each n -cell σ i in Δ , V = ( u 1 , u 2 , … , u τ ) is the set of parameters and X = ( x 1 , x 2 , … , x n ) is the set of variables; σ 1 , σ 2 , … , σ m are all the n -dimensional cells in Δ and Ω = ⋃ i = 1 m σ 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 Δ .
KeywordsPiecewise polynomial, Parametric piecewise polynomial system, Parametric semi-algebraic systems, Discriminant variety, Number of real zeros
ISSN0377-0427
URL http://www.sciencedirect.com/science/article/pii/S037704271100255X
LanguageEnglish
JournalJournal of Computational and Applied Mathematics
Volume236
Number5
Pages924 - 936
Year2011
NoteThe 7th International Conference on Scientific Computing and Applications, June 13–16, 2010, Dalian, China
Edition0
Translation No
Refereed No
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