Some Researches On Piecewise Algebraic Curves And Piecewise Algebraic Varieties

The interpolation of scattered data by multivariate splines is an important topic in computational geometry. Essentially, a key problem on the interpolation by multivariate splines is to study the piecewise algebraic curve and the piecewise algebraic variety forn - dimensional space Rn (n > 2). The piecewise algebraic curve and the piecewise algebraicvariety, as the set of zeros of a bivariate spline function and the set of all common zeros of multivariate splines respectively, are new and important concepts in algebraic geometry and computational geometry. It is obvious that the piecewise algebraic curve (variety) is a kind of generalization of the classical algebraic curve ( variety respectively). The paper applies algebraic geometry, computational geometry, approximation theory to study the following problems: the Nother type theory and the Riemann-Roch type theory of the piecewise algebraic curve; the number of real intersection points of piecewise algebraic curves; the real piecewise algebraic variety and the B-net resultant of polynomials. The following results are obtained.1. Applying multivariate spline theory which was established by Wang RenHong in ([1]>[2]), the Nother theorem of algebraic curve [71] is generalized to the piecewisealgebraic curve. The paper gives the C ? piecewise N(?)ther type theorems on thepartitions of ?l and a star region .2. In this paper, the theory of the so-called "linear series" of sets of places on the piecewise algebraic curve is estabilished and the singular cycle is put into the linear series. The theory of linear series of the piecewise algebraic curve is a kind generalization of that of linear series of the algebraic curve([71]). A complete series of the piecewisealgebraic curve consists of all effective ordinary cycles in an equivalence class {A} andall effective singular cycles which are equivalent specifically to cycle A. By using the theory and multivariate spline theory([1],[2]), the paper discusses and gives theC? piecewise Riemann-Roch type theorems on the partitions of A; and A+. (1): The C? piecewise Riemann-Roch type theorem on the partition ?l: Let p be the genus of an irreducible C? piecewise algebraic curve F with degree m . If i is the index of a complete grn, and there are no fixed places of which center lie onpartition lines, then r - n - p - m(? +1) + i +1.(2) : The C? piecewise Riemann-Roch type theorem on the partition ?+ : Let p be the genus of an irreducible C? piecewise algebraic curve F with degree m. If i is the index of a complete gnr, and there are no fixed places of which center lieon partition lines, then r -n- p- 4m(? +1) + i + 3.3. By using the techniques of an explicit criterion to determine the number of real roots of a univariate polynomial in ([13],[73]); B-net form of bivariate splines function; discriminant sequence of polynomial (cf.[13],[73]) and the number of sign changes in the sequence of coefficients of the highest degree terms of sturm sequence, this paper determines the number of real intersection points two piecewise algebraic curves whose common points are finite. A lower bound of the number of real intersection points is obtained in terms of method of rotation degree of vector field.4. Applying the techniques of real radical ideal, P -radical ideal ( P is a cone ), decomposition of semi-algebraic set in ([72]), affine Hilbert polynomial and B-net formof polynomials on simplex, this paper obtains two theorems of real C? piecewisealgebraic variety dimensions and the real Nullstellensatz in C? spline ring.5. This paper defines the B-net resultant of polynomials, and gives its properties and the relation between it and the solutions of system of equations.