×

zbMATH — the first resource for mathematics

Nöther-type theorem of piecewise algebraic curves on quasi-cross-cut partition. (English) Zbl 1177.14076
Summary: Nöther’s theorem of algebraic curves plays an important role in classical algebraic geometry. As the zero set of a bivariate spline, the piecewise algebraic curve is a generalization of the classical algebraic curve. Nöther-type theorem of piecewise algebraic curves is very important to construct the Lagrange interpolation sets for bivariate spline spaces. In this paper, using the characteristics of quasi-cross-cut partition, properties of bivariate splines and results in algebraic geometry, the Nöther-type theorem of piecewise algebraic curves on the quasi-cross-cut is presented.

MSC:
14H99 Curves in algebraic geometry
65D07 Numerical computation using splines
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Wang R H. The structural characterization and interpolation for multivariate splines. Acta Math Sinica, 18: 91–106 (1975) · Zbl 0358.41004
[2] Wang R H. Multivariate Spline Functions and Their Applications. Beijing/New York: Science Press/Kluwer Academic Publisher, 2001
[3] Wang R H. Multivariate spline and algebraic geometry. J Comput Appl Math, 121: 153–163 (2000) · Zbl 0960.41008 · doi:10.1016/S0377-0427(00)00344-7
[4] Hartshorn R. Algebraic Geometry. New York: Springer Verlag, 1977.
[5] Walker R J. Algebraic Curves. Princeton: Princeton University Press, 1950
[6] Shi X Q, Wang R H. Bezout’s number for piecewise algebraic curves. BIT, 39: 339–349 (1999) · Zbl 0933.65017 · doi:10.1023/A:1022350131468
[7] Wang R H, Xu Z Q. Estimation of the Bezout number for piecewise algebraic curve. Sci China Ser A, 46: 710–717 (2003) · Zbl 1084.65014 · doi:10.1360/02ys0136
[8] Wang R H, Zhu C G. Cayley-Bacharach theorem of piecewise algebraic curves. J Comput Appl Math, 163: 269–276 (2004) · Zbl 1070.14034 · doi:10.1016/j.cam.2003.08.072
[9] Wang R H, Zhu C G. Nöther-type theorem of piecewise algebraic curves. Progr Natur Sci (English Ed), 14: 309–313 (2004) · Zbl 1083.14525 · doi:10.1080/10020070412331343531
[10] Wang R H, Zhu C G. Piecewise algebraic varieties. Progr Natur Sci (English Ed), 14: 568–572 (2004) · Zbl 1083.14526 · doi:10.1080/10020070412331343961
[11] Zhu C G, Wang R H. Piecewise semialgebraic sets. J Comput Math, 23: 503–512 (2005) · Zbl 1084.14058
[12] Zhu C G, Wang R H. Lagrange interpolation by bivariate splines on cross-cut partition. J Comput Appl Math, 195: 326–340 (2006) · Zbl 1097.65023 · doi:10.1016/j.cam.2005.03.090
[13] Zhu C G, Wang R H. Nöther-type theorem of piecewise algebraic curves on triangulation. Sci China Ser A, 50: 1227–1232 (2007) · Zbl 1125.14017 · doi:10.1007/s11425-007-0062-z
[14] Zhu C G, Wang R H. Least-squares fitting of piecewise algebraic curves, Math Probl Eng, Article ID 78702 (2007) · Zbl 1143.65317
[15] Lai Y S. Some researches on piecewise algebraic curves and piecewise algebraic varieties. PhD Dissertation. Dalian: Department of Applied Mathematics, Dalian University of Technology, 2002
[16] Zhu C G. Some researches on piecewise algebraic curves, piecewise algebraic vieriaties and piecewise semialgebraic sets. PhD Dissertation. Dalian: Department of Applied Mathematics, Dalian University of Technology, 2005
[17] Chui C K, Wang R H. Multivariate spline spaces. J Math Anal Appl, 94: 197–221 (1983) · Zbl 0526.41027 · doi:10.1016/0022-247X(83)90014-8
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.