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 Nother type theorems for Cµ 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 Cµ piecewise algebraic curves and the degree, the genus and the smoothness of the curves, hence the Riemann-Roch type theorem of the Cµ piecewise algebraic curve is established.