实分片代数超曲面的连通分支数的上界
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摘要
多元样条是具有一定光滑度的分片多项式,具有一定光滑度的分片代数(超)曲面(即多元样条的零点集)是表示或逼近曲面的重要工具.这篇文章建立了实分片代数超曲面与实分片代数曲线的连通分支数的界.
A multivariate spline is a piecewise polynomial with certain smoothness.Piecewise algebraic(hyper) surfaces with certain smoothness(i.e., the zero set of multivariate splines) are an important tool to represent or approximate surfaces. This paper gives the bounds of the number of connected componects of both real piecewise algebraic hypersurfaces and real piecewise algebraic curves.
A multivariate spline is a piecewise polynomial with certain smoothness.Piecewise algebraic(hyper) surfaces with certain smoothness(i.e., the zero set of multivariate splines) are an important tool to represent or approximate surfaces. This paper gives the bounds of the number of connected componects of both real piecewise algebraic hypersurfaces and real piecewise algebraic curves.
引文
[1]Lai Y S,Wang R H,Wu J M.Solving parametric piecewise polynomial systems.Journal of Computational and Applied Mathematics,2011,236(5):924-936.
[2]Lai Y S,Wang R H,Wu J M.Real zeros of zero-dimensional parametric piecewise algebraic variety.Science China Mathematics,2009,52(4):817-832.
[3]赖义生,段德鑫,方小珂,等.分片代数超曲面的构造与参系数分片多项式系统的研究进展.中国科学:数学,2015,45(9):1423-1440.(Lai Y S,Duan D X,Fang X K,et al.Progress in construction of piecewise algebraic hypersurfaces and solving parametric piecewise polynomial systems.Science China Mathematics,2015,45(9):1423-1440.)
[4]Wang R H.The structural characterization and interpolation for multivariate spline.Acta Math.Sinica,1975,18:91-106.
[5]Wang R H,Shi X Q,Luo Z X,et al.Multivariate Spline Functions and Their Applications.Beijing/New York:Science Press/Kluwer Pub,2001.
[6]Wang R H,Lai Y S.Real piecewise variety.Journal of Computational,2003,21(4):473-480.
[7]Lai Y S,Wang R H.The Noether and Riemann-Roch type theorems for piecewise algebraic curve.Science China Mathematics,2007,52(2):165-182.
[8]Lai Y S,Du W P,Duan D X,et al.Bounds on the number of solutions of polynomial systems and the Betti numbers of real piecewise algebraic hypersurfaces.Journal of Computational and Applied Mathematics,http://dx.doi.org/10.1016/j.cam.2016.11.023.
[9]Farin G.Curves and Surfaces for Computer Aided Geometry Design:A Practical Guide,4th Edition.New York:Academic Press,1997.
[10]Lai Y S,Du W P,Wang R H.The Viro method for construction of piecewise algebraic hypersurfaces.Abstract Applied Analysis,2013,271-290,doi:10.1155/2013/690341.
[11]Li X,Chen F L,Kang H M,et al.A survey on the local refinable splines.Science China Mathematics,2016,59(4):617-644.
[12]Zhu C G,Wang R H.Noether-type theorem of piecewise algebraic curves on quasi-cross-cut partition.Science China Mathematics,2009,52(4):701-708.
[13]Zhu C G,Wang R H.Noether-type theorem of piecewise algebraic curves on triangulation.Science China Mathematics,2007,50(9):1227-1232.
[14]Wang R H,Zhu C G.Cayley-Bacharach theorem of piecewise algebraic curves.Journal of Computational and Applied Mathematics,2004,163(1):269-276.
[15]Lai Y S,Du W P,Wang R H.The Viro method for construction of Bernstein-Bezier algebraic hypersurface piece.Science China Mathematics,2012,55(6):1269-1279.
[16]Khovanskii A G.Fewnomials.Trans.of Math.Monographs,vol.88,AMS,Providence,1991.
[17]Li T Y,Rojas J M,Wang X S.Counting real connected components of trinomial curve intersections and m-nomial hypersurfaces.Discrete Comput.Geom.,2003,30(3):379-414.
[18]Perrucci D.Some bounds for the number of components of real zero sets of sparse polynomials.Discrete Comput.Geom.,2005,34(3):475-495.
[19]Bihan F,Rojas J M,Sottile F.Sharpness of fewnomial bounds and the number of components of a fewnomial hypersurface.Algorithms in Algebraic Geometry,IMA Volumes in Mathematics and Its Applications,Springer,2007,146:15-20.
[2]Lai Y S,Wang R H,Wu J M.Real zeros of zero-dimensional parametric piecewise algebraic variety.Science China Mathematics,2009,52(4):817-832.
[3]赖义生,段德鑫,方小珂,等.分片代数超曲面的构造与参系数分片多项式系统的研究进展.中国科学:数学,2015,45(9):1423-1440.(Lai Y S,Duan D X,Fang X K,et al.Progress in construction of piecewise algebraic hypersurfaces and solving parametric piecewise polynomial systems.Science China Mathematics,2015,45(9):1423-1440.)
[4]Wang R H.The structural characterization and interpolation for multivariate spline.Acta Math.Sinica,1975,18:91-106.
[5]Wang R H,Shi X Q,Luo Z X,et al.Multivariate Spline Functions and Their Applications.Beijing/New York:Science Press/Kluwer Pub,2001.
[6]Wang R H,Lai Y S.Real piecewise variety.Journal of Computational,2003,21(4):473-480.
[7]Lai Y S,Wang R H.The Noether and Riemann-Roch type theorems for piecewise algebraic curve.Science China Mathematics,2007,52(2):165-182.
[8]Lai Y S,Du W P,Duan D X,et al.Bounds on the number of solutions of polynomial systems and the Betti numbers of real piecewise algebraic hypersurfaces.Journal of Computational and Applied Mathematics,http://dx.doi.org/10.1016/j.cam.2016.11.023.
[9]Farin G.Curves and Surfaces for Computer Aided Geometry Design:A Practical Guide,4th Edition.New York:Academic Press,1997.
[10]Lai Y S,Du W P,Wang R H.The Viro method for construction of piecewise algebraic hypersurfaces.Abstract Applied Analysis,2013,271-290,doi:10.1155/2013/690341.
[11]Li X,Chen F L,Kang H M,et al.A survey on the local refinable splines.Science China Mathematics,2016,59(4):617-644.
[12]Zhu C G,Wang R H.Noether-type theorem of piecewise algebraic curves on quasi-cross-cut partition.Science China Mathematics,2009,52(4):701-708.
[13]Zhu C G,Wang R H.Noether-type theorem of piecewise algebraic curves on triangulation.Science China Mathematics,2007,50(9):1227-1232.
[14]Wang R H,Zhu C G.Cayley-Bacharach theorem of piecewise algebraic curves.Journal of Computational and Applied Mathematics,2004,163(1):269-276.
[15]Lai Y S,Du W P,Wang R H.The Viro method for construction of Bernstein-Bezier algebraic hypersurface piece.Science China Mathematics,2012,55(6):1269-1279.
[16]Khovanskii A G.Fewnomials.Trans.of Math.Monographs,vol.88,AMS,Providence,1991.
[17]Li T Y,Rojas J M,Wang X S.Counting real connected components of trinomial curve intersections and m-nomial hypersurfaces.Discrete Comput.Geom.,2003,30(3):379-414.
[18]Perrucci D.Some bounds for the number of components of real zero sets of sparse polynomials.Discrete Comput.Geom.,2005,34(3):475-495.
[19]Bihan F,Rojas J M,Sottile F.Sharpness of fewnomial bounds and the number of components of a fewnomial hypersurface.Algorithms in Algebraic Geometry,IMA Volumes in Mathematics and Its Applications,Springer,2007,146:15-20.