分片代数曲线与分片代数簇的若干研究
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摘要
利用多元样条函数进行散乱数据插值是计算几何中一个非常重要的课题。本质上，解决多元样条函数空间的插值结点的适定性问题关键在于研究分片代数曲线，在高维空间里就是研究分片代数簇。
分片代数曲线作为二元样条函数的零点集合，分片代数簇作为一些多元样条函数的公共零点集合，它们是代数几何与计算几何中一种新的重要概念，显然也是经典代数曲线与代数簇的推广。本文应用代数几何，计算几何，函数逼近论等学科的基本理论，分别就分片代数曲线的N(?)ther型与Riemann-Roch型定理；分片代数曲线的实交点数；实分片代数簇以及多项式的B-网结式进行研究。主要工作如下：
1：利用王仁宏在文献([1]，[2])中关于多元样条函数的基本理论，本文将代数曲线的N(?)ther定理([71])推广到分片代数曲线上，给出了剖分为Δ_1和星形剖分上的C~μ分片N(?)ther型定理。
2：本文将奇异循环纳入分片代数曲线的线性列中，建立了由分片代数曲线的支组所构成的“线性列”理论，这一理论是相应代数曲线“线性列”理论([71])的推广。分片代数曲线的完全列是由所有属于同一等价类的有效正常循环以及所有与它特等价的有效奇异循环构成。利用这一理论以及多元样条函数的基本理论([1]，[2])，本文着重讨论并给出了剖分为Δ_1和Δ_+的C~μ分片Riemann-Roch型定理。
Δ_1上的C~μ分片Riemann-Roch型定理：
设p是次数为们的不可约C~μ分片代数曲线F的亏格。若i为完全列g_n~r的指标，且列g_n~r中不存在中心落在剖分线上的不动支，则r=n-p-m(μ+1)+i+1。
Δ_+上的C~μ分片Riemann-Roch型定理：
设p是次数为m的不可约C~μ分片代数曲线F的亏格。若为完全列g_n~r的指标，且列g_n~r不存在中心在剖分线上的不动支， 则r=n-p-4m(μ+1)+i+3。
大连理工大学博士论义：分片代数价线／I片代数簇的若十M穴
3：利用杨路，张景中，侯晓荣在文献V13］，［73］）中关于一元多项式实根的显式
判准，以及二元样条函数的B－网形式，多项式的判别序列和 Sturm序列的最高次数项系
数序列的变号数，本文给出了两个分片代数曲线的丈交点数（假设公共点是有限的）的
计算公式。使用向量场的旋转度方法，也给出了交。l、）数的一个下界计算公上巳
4：应用多项式在单纯形上的B－网形式以及文献门21）中的实根理想，锥根理想，
半代数簇分解定理，本文得出了实C”分片代数簇的＿二个维数定理和C。‘样条空间的实零
点定理（Real Nullstellensatz）。
5：本文定义了多项式r网结式，导出了卜；叫结式的性质和分网结人与方程组的解
之间的关系。
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 for
n - dimensional space Rn (n > 2). The piecewise algebraic curve and the piecewise algebraic
variety, 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 piecewise
algebraic curve. The paper gives the C ? piecewise N(?)ther type theorems on the
partitions 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 piecewise
algebraic curve consists of all effective ordinary cycles in an equivalence class {A} and
all 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 the
C? 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 on
partition 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 lie
on 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 form
of polynomials on simplex, this paper obtains two theorems of real C? piecewise
algebraic 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.
分片代数曲线作为二元样条函数的零点集合，分片代数簇作为一些多元样条函数的公共零点集合，它们是代数几何与计算几何中一种新的重要概念，显然也是经典代数曲线与代数簇的推广。本文应用代数几何，计算几何，函数逼近论等学科的基本理论，分别就分片代数曲线的N(?)ther型与Riemann-Roch型定理；分片代数曲线的实交点数；实分片代数簇以及多项式的B-网结式进行研究。主要工作如下：
1：利用王仁宏在文献([1]，[2])中关于多元样条函数的基本理论，本文将代数曲线的N(?)ther定理([71])推广到分片代数曲线上，给出了剖分为Δ_1和星形剖分上的C~μ分片N(?)ther型定理。
2：本文将奇异循环纳入分片代数曲线的线性列中，建立了由分片代数曲线的支组所构成的“线性列”理论，这一理论是相应代数曲线“线性列”理论([71])的推广。分片代数曲线的完全列是由所有属于同一等价类的有效正常循环以及所有与它特等价的有效奇异循环构成。利用这一理论以及多元样条函数的基本理论([1]，[2])，本文着重讨论并给出了剖分为Δ_1和Δ_+的C~μ分片Riemann-Roch型定理。
Δ_1上的C~μ分片Riemann-Roch型定理：
设p是次数为们的不可约C~μ分片代数曲线F的亏格。若i为完全列g_n~r的指标，且列g_n~r中不存在中心落在剖分线上的不动支，则r=n-p-m(μ+1)+i+1。
Δ_+上的C~μ分片Riemann-Roch型定理：
设p是次数为m的不可约C~μ分片代数曲线F的亏格。若为完全列g_n~r的指标，且列g_n~r不存在中心在剖分线上的不动支， 则r=n-p-4m(μ+1)+i+3。
大连理工大学博士论义：分片代数价线／I片代数簇的若十M穴
3：利用杨路，张景中，侯晓荣在文献V13］，［73］）中关于一元多项式实根的显式
判准，以及二元样条函数的B－网形式，多项式的判别序列和 Sturm序列的最高次数项系
数序列的变号数，本文给出了两个分片代数曲线的丈交点数（假设公共点是有限的）的
计算公式。使用向量场的旋转度方法，也给出了交。l、）数的一个下界计算公上巳
4：应用多项式在单纯形上的B－网形式以及文献门21）中的实根理想，锥根理想，
半代数簇分解定理，本文得出了实C”分片代数簇的＿二个维数定理和C。‘样条空间的实零
点定理（Real Nullstellensatz）。
5：本文定义了多项式r网结式，导出了卜；叫结式的性质和分网结人与方程组的解
之间的关系。
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 for
n - dimensional space Rn (n > 2). The piecewise algebraic curve and the piecewise algebraic
variety, 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 piecewise
algebraic curve. The paper gives the C ? piecewise N(?)ther type theorems on the
partitions 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 piecewise
algebraic curve consists of all effective ordinary cycles in an equivalence class {A} and
all 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 the
C? 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 on
partition 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 lie
on 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 form
of polynomials on simplex, this paper obtains two theorems of real C? piecewise
algebraic 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.
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80Bernstein,D.N.,The Number of a system of equations,Functional Anal.Appl.,9(3) (1975) ,1-4．
81C.L.Bajaj,Guoliang Xu,Piecewise rational approximations of real algebraic curves,J.Comput.Math.,Vol.15,No.1,1997,55-71．
82C.L.Bajaj,Guoliang Xu,A-splines: Local interpolation and approximation using
Gk-continueus piecewise real algebraic curves,CAGD,16(1999) ,557-578．
83S.S .Abhyankar,C.L.Bajaj,Automatics parameterization of rational curves and surface I: conies and conicoids,CAGD ,19(1) (1987a),11-14． .
84S.S .Abhyankar,C.L.Bajaj, Automatics parameterization of rational curves and surface II: cobics and cobicoids,CAGD, 19(9) (1988) 499-502．
85S.S .Abhyankar,C.L.Bajaj,Automatics parameterization of rational curves and surface III: algebraic plane curves,CAGD ,5 (1988) ,309-321．
86L.Blum,M.Shub and S.Smale,On a theory of computation and complexity over the real numbers: NP-completeness,recursive functions,and universal machines,Bull.Amer.Math.Soc.,2(1989) ,1-46．
87J.F.Canny and I.Emiris,An efficient algorithm for the sparse mixed resultant,in: Proc.AAECC,Puerto Rico,Lecture Notes in Computer Science,Vol.263(Springer,Berlin,1993) ,89-104．
88W.Fulton,Intersection Theory ,Springer,Berlin,1984．
89W.Fulton,Introduction to Toric Varieties,Annals of Mathematics Studies, Vol.131(Princeton University Press,Princeton,NJ,1993) .
90 I.M.Gel'fand,M.M.Kapranov and A.V.Zelevinsky,Discriminants of polynomials in several variables and triangulations of Newton polytopes,Algebra and Anal.,2(1990) ,1-62 (translated from Rusian).
91A.G.Khovanskii,Newton polyhedra and the genus of complete intersections, Functional Anal.Appl.,12(1) (1978) 51-61．
92A.G.Kushnirenko,Newton polytopes and the Bezout theorem,Functional Anal.Appl.,10(3) (1976) , 82-83(translated from Russian).
93T.Y.Li,T.Sauer and J.A.Yorke,Numerical solution of a class of deficient polynomial systems,SIAMJ.Numer.Anal.,24(2) (1987) .
94T.Oda,Convex bodies and algebraic geometry: an introduction to the Theory of Toric Varieties ,Springer,Berlin,1988．
95P.Pedersen and B.Sturmfels,Product formulas for sparse resultants and chow forms, Math.Z.,214(1993) ,377-396．
96B.Sturmfels,On the Newton polytope of the resultant,J.Algebraic Combinatorics, 3(1994) ,207-236．
97B.Sturmfels and A.Zelevinsky,Multigraded resultants of Sylvester type,J.Algebra, 163(1994) , 115-127．
98J.Verschelde and A.Haegemans,Homotopies for solving polynomial systems within a bounded domain,Theoret.Comput.Sci.,133(1994) , 141-161．
99C.W.Wampler,Bezout number calculations for multi-homogeneous polynomial systems,App.Math.Comput.,51(1992) ,143-157．
100J.Maurice Rojas,A convex geometric approach to counting the roots of a polynomial system,Theoretical Computer Science,133(1994) ,105-140．
101L.L.Schumaker,On the dimension of spaces of piecewise polynomials in two
variables,in "Multivariate Approximation Theory",W.Schempp and K.Zeller (eds.),pp.396-412,Birkhauser,Basel,1979．
102H.B.Curry and I.J.Schoenberg,On P61ya frequency functions and their limits,J.d'Analyse Math.17(1966) ,71-107．
103C.de Boor,Topices in multivariate approximation theory,in "Topices in Numerical Analysis",P.R.Turner (ed.),Lecture Notes Mathematices,Springer-Verlag,965(1982) ,39-78．
104W.Dahmen and C.A.Micchelli ,Recent progress in multivariate splines,interpolating cardinal splines as their degree rends to infinity,Israel J.Ward (eds.),Academic Press,1983,27-121．
105 贾荣庆，箱样条研究的新进展，高校应用数学学报,2(1987) ,330-343．
106C.A.Micchelli,A constructive approach to Kergin interpolation in Rk: Multivariate B-splines and Lagange interpolation,Rocky Mountion J.Math.,10(1980) ,485-497．
107C.de Boor and R.DeVore,Approximation by smooth multivariate splines,Trans.Amer.Math.Soc.,276(1983) ,775-788．
108W.Dahmen,On multivariate B-splines,SIAM J.Numer.Anal.,17(1980) ,179-191．
109K.Hollig,Box splines,in "Approximation Theorey",V.C.K.Chui,L.L .Schumaker and J.Ward (eds.),Academic Press,New York,1986,71-95．
110C.de Boor and K.Hollig,B-splines from parallepipeds.J.d'Anal.Math.,41(1982/1983) ,99-115．
111C.A.Micchelli,On a numerically efficient method for computing multivariate B-splines,in "Multivariate Approximation Theory".W.Schempp and K.Zeller (eds.),Birkhauser,Basel,1979,211-248．
112G.G.Lorentz,Bernstein Polynommials,Torento,1953．
113C.de Boor,B-form basics,in "Geometric Modelling:,G.Fatin(eds.) SIAM ,Philadelphia,131-148．
114G.Farin,Triangular Bernstein-Bezier patches,CAGD,3(1986) ,83-127．
115 郭竹瑞，贾荣庆，多元样条研究中的B-网方法，数学进展,19(1990) ,189-198．
116G.Farin,Bezier polynomials over triangles and the construction of piecewise Cr polynomials,TR/91,Dept.of Math.,Brunei Univ.,Uxbridge,Middlesex,U.K.,1980．
117G.Chang and P.J.Davis,The convexity of Bernstein polynomials over triangles,J.Approx.Th.,40(1984) ,11-28．
118G.Chang and Y.Feng,An improved condition for the convexity of Bernstein-Bezier surfaces over triangles,CAGD,1(1984) ,279-283．
119G.Chang and Y.Feng,A new proof for the convexity of the Bernstein-Bezier surfaces over triangles,ChineseAnn.Math.,6B(1989) ,173-176．
120X.Q.shi,The dimensions of spline spaces and their singularity,J.Comp.Math.,10(1992) ,224-230．
121D.Hong,Spaces of bivariate spline functions over triangulation,Approx Theory Appl.,7(1991) ,56-75．
122X.Q.Shi,The dimensions of spline spaces Chinese Science and Bulletin,37(1992) ,436-437．
123R.H.Wang and X.Q.Shi,A kind of cubic Cl interpolations in the n-dimensional finite element.J.Math.Res.and Exp.,9(1989) ,173-179．
124X.Q.Shi and R.H.Wang,The existence conditions of space S12(n) ,Chinese Science and Bulletin,34(1989) ,2015．
125P.Alfeld,B.Piper and L.L.Schumaker,Spaces of bivariate splines on triangulations with holes,Approx.Theory Appl.,3(1987) ,1-10．
126Z.X.Luo and R.H.Wang,Structure and application of algebraic spline curve and surface,J.Math.Res.&Exp.,12(1992) ,579-582．
2 王仁宏，多元样条函数及其应用，纯粹数学与应用数学专著(第28号)，科学出版社，1994．
3 王仁宏，数值逼近，高等教育出版社，1999．
4 王仁宏，梁学章，多元函数逼近，科学出版社，1990．
5 王仁宏，任意剖分下的多元样条分析，中国科学，数学专辑Ⅰ，1979，215-226．
6 王仁宏，任意剖分下的多元样条分析(Ⅱ)-空间形式，高等学校计算数学学报，1 (1980)，78-81．
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9 王仁宏，何天晓，拟贯穿剖分下多元样条空间基，中国科学，A辑，1 (1986)，19-25．
10 王仁宏，关于多元样条空间的维数，科学通报，6(1988)，473-474．
11 王仁宏，何天晓，不均匀第二型三角剖分下的带有边界条件的样条函数空间，科学通报，4 (1985)，249-251．
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19 莫宗坚，代数学，北京大学出版社，1986．
20 施锡泉，高维样条，吉林大学博士论文，1988．
21 罗钟铉，非线性样条函数研究，大连理工大学博士论文，1992．
22 苏志勋，分片代数曲线曲面及其在CAGD中的应用，大连理工大学博士论文，1993．
23 赵国辉，多元样条中的几个相关问题研究，大连理工大学博士论文，1996．
24 刘秀平，分片代数曲线问题的研究，大连理工大学博士论文，1999．
25 孙家昶，可三向剖分上S_3~1空间的B样条对偶基与拟插值，应用数学学报，14(1991)，470-477．
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27 C. K. Chui and R. H. Wang, Multivariate spline spaces, J. Math. Anal. Appl., 94(1983), 197-221.
28 C. K. Chui and R. H. Wang, On smooth multivariate spline functions, Math. Comp., 41(1983), 131-142.
29C.K.Chui and R.H.Wang,Bases of bivariate spline spaces with crosscut grid partitions,数学研究与评论.1(1982) ,1-3．
30C.K.Chui and R.H.Wang,A generalization of univariate splines with equally spaced knots to multivariate splines,ibid,2(1982) ,99-104．
31C.K.Chui and R.H.Wang,Space of bivariate cubic and quartic splines on type-1 triangulations,J.Math.Anal.Appl.,101(1984) ,540-554．
32C.K.Chui and R.H.Wang,Bivariate cubic B-splines relative to crosscut triangulation,Chin.Ann.Math.,4B(4) (1983) ,509-523．
33C.K.Chui,T.X.He and R.H.Wang,The C2 quartic splines spaces on a four-directional mesh,Approx.Theory Appl.,3:4(1987) ,32-36．
34C.K.Chui and R.H.Wang,Bivariate B-splines on triangulated rectangles,in "Approximation Theory IV",C.K.Chui,L.L.Schumaker and J.D.Ward(eds.),Acad.Press,New York,1983,413-418．
35C.K.Chui and R.H.Wang,Concerning C1 B-splines on triangulations of non-uniform rectangular partition, Approx.Theory Appl., 1:1(1984) ,11-18．
36C.K.Chui and R.H.Wang,Multivarate B-splines on triangulated rectangles,J.Math.Anal.Appl., 92(1983) ,533-551．
37C.K.Chui,L.L.Schumaker and R.H.Wang,On spaces of piecewise polynomials with boundary conditions II.Type-1 triangulations,Canadian Mathematical Society Conf.Proceedings,3(1983) ,51-56．
38C.K.Chui,L.L.Schumaker and R.H.Wang, On spaces of piecewise polynomials with boundary conditions III.Type-2 triangulations,Canadian Mathematical Society Conf.Proceedings,3(1983) ,67-80．
39Y.S.Chou,L.Y.Su and R.H.Wang,The dimensions of bivariate spline space over triangulations , in "Multivariate Approximation Theory III,"W.Schempp&K.Zeller(eds.),Birkhauser,Basel,1985,71-83．
40R.H.Wang,T.X.He, X.Y.Liu and S.C.Wang,An intergral method for constructing bivariate spline functions,J.Comp.Math.,7(1989) ,244-261．
41R.H.Wang,Multivariate spline and algebraic geometry,J.Comp.Appl.Math., 121(2000) ,153-163．
42X.Q.Shi and R.H.Wang,Bezout number for piecewise algebraic curves,BIT, 2(1999) ,339-349．
43R.H.Wang,G.H.Zhao,An introduction to piecewise algebraic curves,in:T.Mitsui(Ed),Theory and Application of Scientific and Technical Computing,RIMS,Kyoto University,(1997) ,196-205．
44R.H.Wang,On piecewise algebraic curves,in:F.Fontanella,etal.,(Eds.),Advanced Topics in Multivariate Approximation,World scientific, Sirgapore,1996,pp.355-361．
45L.L.Schumaker,Bounds on the dimension of spaces of multivariate piecewise polynomials,Rocky Mountain J.Math.,14(1984) , 251-264．
46L.L.Schumaker,On supper splines and finite elements,SIAMJ.Numer.Anal., 4(1989) ,997-1005．
47O.Davydov,G.Nurnberger,Interpolation by C' splines of degree q>4 on triangulations,J.Comp.Appl.Math.,126(2000) ,159-183．
48O.Davydov,G.NUrnberger,F.Zeilfelder,Interpolation by cubic splines on triangulations,in:C.K.Chui,L.L.Schumaker(Eds.), Approximation Theory IX,Vanderbilt University Press,Nashville,1998,pp.17-24．
49O.Davydov,G.NUrnberger,F.Zeilfelder,Approximation order of bivariate spline interpolation for arbitrary smoothness,J.Comp.Appl.Maths.,90(1998) ,117-134．
50O.Davydov,G.NUrnberger,F.Zeilfelder,Interpolation by splines on triangulations, in: M.W.Muller,M.D.Buhmann,D.H.Mache,M.Felten(Eds.),New Developments in Approximation Theory.International Series of Numerical Mathematics,Vol.132,Birkhauser,Basel,1999,pp.49-70．
51O.Davydov,G.NUrnberger,F.Zeilfelder,Cubic spline interpolation on nested polygon triangulations,in: A.Cohen,C.Rabut and L.L.Schumaker(Eds.),Curves and Surfaces,st.Malo,1999． Vanderbilt University Press,to appear.
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55G.NUrnberger,T.Riefiinger,Bivariate spline interpolation at grid points,Numer.Math.,71(1995) ,91-119．
56G.NUrnberger,G.Walz,Error analysis in interpolation by bivariate C1-splines, IMAJ.Numer.Anal.,18(1998) ,485-508．
57G.NUrnberger,F.Zeilfelder,Interpolation by spline spaces on classes of triangulations,J.Comp.Appl.Math.,119(2000) ,347-376．
58G.NUrnberger,F.Zeilfelder, On bivariate spline spaces,in: W.HauBmann, K.Jetter and M.Reimer(Eds.),Advances in Multivariate Approximation,Wiley,1998,pp.227-231．
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61O.Davydov, L.L.Schumaker,Locally linearly independent bases for bivariate
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62O.Davydov, M.Sommer,H.Strauss,On almost interpolation by multivariate splines,in: g.Nurnberger,J.W.Schmidt,G.Walz(Eds.),Multivariate Approximation and Splines,International Series of Numerical Mathematics,Vol.125,Birkhauser,Basel,1997,pp.45-58．
63O.Davydov,M.Sommer,H.Strauss, Locally linearly independent systems and almost interpolation,in: G.NUrnberger, J.W.Schmidt,G.Walz(Eds.),Multivariate Approximation and Splines , International Series of Numerical Mathematics,Vol.125,Birkhauser,Basel,1997,pp.59-72．
64O.Davydov,M.Sommer,H.Strauss,On almost interpolation and locally linearly independent basis,East J.Approx.,5(1999) ,67-88．
65O.Davydov,M.Sommer,H.Strauss,Interpolation by bivariate linear splines, J.Comp.Appl.Math.,119(2000) ,115-134．
66M.Sommer,H.Strauss,A condition of Schoenberg-Whitney type for multivariate spline interpolation,Adv.Comput.Math.,5(1996) ,381-397．
67Ren-Hong Wang,Yi-sheng Lai,Piecewise algebraic curves,J.Comput.Appl.Math., to appear.
68Ren-Hong Wang , Yi-sheng Lai , Real piecewise algebraic variety , J.Comput.Math.,to appear.
69Ren-Hong Wang,Yi-sheng Lai,Piecewise algebraic variety,Proceedings of The 5th China-Japan Seminar on Numerical Mathematics,2000．
70Ren-hong Wang,Multivariate spline and its applications in science and technology, Rend.Sem.Mat.Fis.Milano,LXIII(1993) ,213-229．
71R.J.Walker,Algebraic Curves,Princeton University Press.Princeton,NJ,1950．
72Jacek Bochnak,Michel coste,and Marie-Francoise Roy,Real Algebraic Geometry, Spinger,1998．
73YangLu,Hou Xiaorong,Zeng Z.B,A complete discrimination system for polynomials, Science in China,SerE,1996,39(6) : 626-646．
74David Cox,John Little,and Dond O'shea,Ideals,Varieties,and Algorithms, Springer-Verlag,1992．
75David Cox,John Little,and Dond O'shea,Using Algebraic Geometry.Springer, 1998．
76G.Stengle,A Nullstellensatz and a positivstellsatz in semialgebraic geometry, Math.Ann.,207(1994) ,87-97．
77I.N.Shafarevich,Basic Algebraic Geometry,Springer,1980．
78 张锦炎，常微分议程几何理论与分支问题，北京大学出版社,1987．
79R.Hartshone,Algebraic Geometry,Springer,1970．
80Bernstein,D.N.,The Number of a system of equations,Functional Anal.Appl.,9(3) (1975) ,1-4．
81C.L.Bajaj,Guoliang Xu,Piecewise rational approximations of real algebraic curves,J.Comput.Math.,Vol.15,No.1,1997,55-71．
82C.L.Bajaj,Guoliang Xu,A-splines: Local interpolation and approximation using
Gk-continueus piecewise real algebraic curves,CAGD,16(1999) ,557-578．
83S.S .Abhyankar,C.L.Bajaj,Automatics parameterization of rational curves and surface I: conies and conicoids,CAGD ,19(1) (1987a),11-14． .
84S.S .Abhyankar,C.L.Bajaj, Automatics parameterization of rational curves and surface II: cobics and cobicoids,CAGD, 19(9) (1988) 499-502．
85S.S .Abhyankar,C.L.Bajaj,Automatics parameterization of rational curves and surface III: algebraic plane curves,CAGD ,5 (1988) ,309-321．
86L.Blum,M.Shub and S.Smale,On a theory of computation and complexity over the real numbers: NP-completeness,recursive functions,and universal machines,Bull.Amer.Math.Soc.,2(1989) ,1-46．
87J.F.Canny and I.Emiris,An efficient algorithm for the sparse mixed resultant,in: Proc.AAECC,Puerto Rico,Lecture Notes in Computer Science,Vol.263(Springer,Berlin,1993) ,89-104．
88W.Fulton,Intersection Theory ,Springer,Berlin,1984．
89W.Fulton,Introduction to Toric Varieties,Annals of Mathematics Studies, Vol.131(Princeton University Press,Princeton,NJ,1993) .
90 I.M.Gel'fand,M.M.Kapranov and A.V.Zelevinsky,Discriminants of polynomials in several variables and triangulations of Newton polytopes,Algebra and Anal.,2(1990) ,1-62 (translated from Rusian).
91A.G.Khovanskii,Newton polyhedra and the genus of complete intersections, Functional Anal.Appl.,12(1) (1978) 51-61．
92A.G.Kushnirenko,Newton polytopes and the Bezout theorem,Functional Anal.Appl.,10(3) (1976) , 82-83(translated from Russian).
93T.Y.Li,T.Sauer and J.A.Yorke,Numerical solution of a class of deficient polynomial systems,SIAMJ.Numer.Anal.,24(2) (1987) .
94T.Oda,Convex bodies and algebraic geometry: an introduction to the Theory of Toric Varieties ,Springer,Berlin,1988．
95P.Pedersen and B.Sturmfels,Product formulas for sparse resultants and chow forms, Math.Z.,214(1993) ,377-396．
96B.Sturmfels,On the Newton polytope of the resultant,J.Algebraic Combinatorics, 3(1994) ,207-236．
97B.Sturmfels and A.Zelevinsky,Multigraded resultants of Sylvester type,J.Algebra, 163(1994) , 115-127．
98J.Verschelde and A.Haegemans,Homotopies for solving polynomial systems within a bounded domain,Theoret.Comput.Sci.,133(1994) , 141-161．
99C.W.Wampler,Bezout number calculations for multi-homogeneous polynomial systems,App.Math.Comput.,51(1992) ,143-157．
100J.Maurice Rojas,A convex geometric approach to counting the roots of a polynomial system,Theoretical Computer Science,133(1994) ,105-140．
101L.L.Schumaker,On the dimension of spaces of piecewise polynomials in two
variables,in "Multivariate Approximation Theory",W.Schempp and K.Zeller (eds.),pp.396-412,Birkhauser,Basel,1979．
102H.B.Curry and I.J.Schoenberg,On P61ya frequency functions and their limits,J.d'Analyse Math.17(1966) ,71-107．
103C.de Boor,Topices in multivariate approximation theory,in "Topices in Numerical Analysis",P.R.Turner (ed.),Lecture Notes Mathematices,Springer-Verlag,965(1982) ,39-78．
104W.Dahmen and C.A.Micchelli ,Recent progress in multivariate splines,interpolating cardinal splines as their degree rends to infinity,Israel J.Ward (eds.),Academic Press,1983,27-121．
105 贾荣庆，箱样条研究的新进展，高校应用数学学报,2(1987) ,330-343．
106C.A.Micchelli,A constructive approach to Kergin interpolation in Rk: Multivariate B-splines and Lagange interpolation,Rocky Mountion J.Math.,10(1980) ,485-497．
107C.de Boor and R.DeVore,Approximation by smooth multivariate splines,Trans.Amer.Math.Soc.,276(1983) ,775-788．
108W.Dahmen,On multivariate B-splines,SIAM J.Numer.Anal.,17(1980) ,179-191．
109K.Hollig,Box splines,in "Approximation Theorey",V.C.K.Chui,L.L .Schumaker and J.Ward (eds.),Academic Press,New York,1986,71-95．
110C.de Boor and K.Hollig,B-splines from parallepipeds.J.d'Anal.Math.,41(1982/1983) ,99-115．
111C.A.Micchelli,On a numerically efficient method for computing multivariate B-splines,in "Multivariate Approximation Theory".W.Schempp and K.Zeller (eds.),Birkhauser,Basel,1979,211-248．
112G.G.Lorentz,Bernstein Polynommials,Torento,1953．
113C.de Boor,B-form basics,in "Geometric Modelling:,G.Fatin(eds.) SIAM ,Philadelphia,131-148．
114G.Farin,Triangular Bernstein-Bezier patches,CAGD,3(1986) ,83-127．
115 郭竹瑞，贾荣庆，多元样条研究中的B-网方法，数学进展,19(1990) ,189-198．
116G.Farin,Bezier polynomials over triangles and the construction of piecewise Cr polynomials,TR/91,Dept.of Math.,Brunei Univ.,Uxbridge,Middlesex,U.K.,1980．
117G.Chang and P.J.Davis,The convexity of Bernstein polynomials over triangles,J.Approx.Th.,40(1984) ,11-28．
118G.Chang and Y.Feng,An improved condition for the convexity of Bernstein-Bezier surfaces over triangles,CAGD,1(1984) ,279-283．
119G.Chang and Y.Feng,A new proof for the convexity of the Bernstein-Bezier surfaces over triangles,ChineseAnn.Math.,6B(1989) ,173-176．
120X.Q.shi,The dimensions of spline spaces and their singularity,J.Comp.Math.,10(1992) ,224-230．
121D.Hong,Spaces of bivariate spline functions over triangulation,Approx Theory Appl.,7(1991) ,56-75．
122X.Q.Shi,The dimensions of spline spaces Chinese Science and Bulletin,37(1992) ,436-437．
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