The Nöther and Riemann-Roch type theorems for piecewise algebraic curve

Abstract

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, the Nöther type theorems for C μ piecewise algebraic curves are obtained. The theory of the linear series of sets of places on the piecewise algebraic curve is also established. In this theory, singular cycles are put into the linear series, and a complete series of the piecewise algebraic curves consists of all effective ordinary cycles in an equivalence class and all effective singular cycles which are equivalent specifically to any effective ordinary cycle in the equivalence class. This theory is a generalization of that of linear series of the algebraic curve. With this theory and the fundamental theory of multivariate splines on smoothing cofactors and global conformality conditions, and the results on the general expression of multivariate splines, we get a formula on the index, the order and the dimension of a complete series of the irreducible C μ piecewise algebraic curves and the degree, the genus and the smoothness of the curves, hence the Riemann-Roch type theorem of the C μ piecewise algebraic curve is established.

Real zeros of the zero-dimensional parametric piecewise algebraic variety

“…, σ m be a given fixed ordering of the n-cells in Δ, and let Ω = m i=1 σ i . Now, we recall the definitions of C r (Δ) and C r k (Δ) in [2][3][4][5][6][7]. Definition 1.1.…”

“…It is clear that C r (Δ) and C r k (Δ) are a Noether ring and a finite dimensional linear vector space, respectively, called a C r spline ring and a multivariate spline space with degree k and smoothness r (see [2,5,7]), respectively. Z is called a piecewise algebraic variety [2] if there exist The piecewise algebraic variety is a new and important topic of computational geometry and algebraic geometry.…”

“…Piecewise algebraic varieties have many applications in various scientific, academic as well as industrial-domains, such as CAD, CAM, CAE and image processing [5] . Some properties of the piecewise algebraic variety were discussed in [2,7], for instance, theorems on the dimension of the real piecewise algebraic variety and the real Nullstellensatz in C μ spring ring had been obtained in [2], the Nöther's type theorem and the Riemann-Roch type theorem for piecewise algebraic curve were established in [7].…”

A nearest neighbor sweep circle algorithm for computing discrete Voronoi tessellations

“…Voronoi tessellations (VT's) are fundamental geometric data structures that appear naturally in a broad range of scientific areas including biology, statistics and computer science. For excellent reviews of the theory and application of VT's and similar domains see [1,5,7,8].…”