Real zeros of the zero-dimensional parametric piecewise algebraic variety

Abstract

The piecewise algebraic variety is the set of all common zeros of multivariate splines.We show that solving a parametric piecewise algebraic variety amounts to solve a finite number of parametric polynomial systems containing strict inequalities. With the regular decomposition of semialgebraic systems and the partial cylindrical algebraic decomposition method, we give a method to compute the supremum of the number of torsion-free real zeros of a given zero-dimensional parametric piecewise algebraic variety, and to get distributions of the number of real zeros in every n-dimensional cell when the number reaches the supremum. This method also produces corresponding necessary and sufficient conditions for reaching the supremum and its distributions. We also present an algorithm to produce a necessary and sufficient condition for a given zero-dimensional parametric piecewise algebraic variety to have a given number of distinct torsion-free real zeros in every n-cell in the n-complex.

Real root classification of parametric spline functions

“…Thus, it is of theoretic and practical significance to study the counting and isolating the real roots of spline functions and its related problems. There exists several work on this issue [6][7][8][9][10][11][12][13]. For univariate case, Goodman [6] and de Boor [7] studied the relationship between the number of real roots of a univariate spline and the sequence of its B-spline coefficients, which provides new bounds on the number of real roots of the spline function.…”

“…In 2008, Wang and Wu [10] proposed an algorithm to isolate real roots of a given univariate spline based on the use of Descartes rule of signs with its B-spline coefficients and de Casteljau algorithm. For multivariate case, Lai et al [11,12] gave the method to compute the supremum and its distribution of the distinct torsion-free real zeros of a given parametric piecewise polynomial system. In 2011, Wu and Zhang [13] presented an algo-S n ½x 0 ; x 1 ; .…”

“…From the RRC of SðxÞ, we can obtain the conditions on b; c; d such that sðxÞ P 0; 8x 2 ðÀ1; 1Þ, i.e., case (3), case (9) or case (11) holds. Meanwhile, we also found the maximum of the number of the real roots of SðxÞ is 3 (counted with multiplicity) if and only if one of cases (2), (4) (7) and (8) hold.…”

The Viro Method for Construction of Piecewise Algebraic Hypersurfaces

“…. , = in Δ such that each is -dimensional, each contains , and and +1 are adjacent for each (see [1,2]).…”

“…, be a given, fixed, ordering of the -cells in Δ, and let Ω = ⋃ =1 . Now, we recall the definitions of (Δ) and (Δ) (see [1,2]). Definition 1.…”