Solving parametric piecewise polynomial systems

[摘要] We deal with C-r smooth continuity conditions for piecewise polynomial functions on A, where Delta is an algebraic hypersurface partition of a domain Q in R-n. Piecewise polynomial functions of degree, at most, k on Delta 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, ... , f(n)) = {X is an element of Omega vertical bar f(1)(V, X) = 0, ... , fn(V, X) = 0}, where fw is an element of Ck(omega)r(omega) (Delta) and f(omega) vertical bar sigma(i) is an element of Q vertical bar V vertical bar vertical bar X vertical bar for each n-cell sigma(i) in Delta, V = (u(1), u(2), ... , u(r)) is the set of parameters and X = (x(1), x(2), ... , x(n)) is the set of variables: sigma(1), sigma(2), ... , sigma(m) are all the n-dimensional cells in Delta and Omega = U-i=1, sigma(1). 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 Delta. (C) 2011 Elsevier B.V. All rights reserved.

[发布日期] 2011-10-01

[效力级别]  Proceedings Paper

[关键词] Piecewise polynomial;Parametric piecewise polynomial system;Parametric semi-algebraic systems;Discriminant variety;Number of real zeros