Cayley-Bacharach theorem of piecewise algebraic curves. (English) Zbl 1070.14034

Wang, Renhong; Zhu, Chungang

J. Comput. Appl. Math. 163, No. 1, 269-276 (2004).

A piecewise algebraic curve is a generalization of the classical algebraic curve. The piecewise algebraic curve is not only important for the interpolation by bivariate splines but also a useful tool for studying traditional algebraic curves.
It is well known that the Cayley-Bacharach theorem is an important and classical result in algebraic geometry. This result has been successively generalized, improved and reinterpreted, and this development continues today [see for instance the paper by S.-L. Tan, J. Algebr. Geom. 9, No. 2, 201–222 (2000; Zbl 0953.14033)].
In this paper, by using Bézout’s theorem and a Noether-type theorem of piecewise algebraic curves, the Cayley-Bacharach theorem and Hilbert function of \(C^0\) piecewise algebraic curves are presented.

Reviewer: Juan Rafael Sendra (Alcalá de Henares)

MSC:

14H50	Plane and space curves
14F05	Sheaves, derived categories of sheaves, etc. (MSC2010)
65D07	Numerical computation using splines
13D40	Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series

Keywords:

bivariate spline functions; piecewise algebraic curves; Cayley-Bacharach theorem; Bézout’s theorem

Citations:

Zbl 0953.14033

References:

[1]	L.H. Cui, Multivariate Lagrange interpolation and multivariate Kergin interpolation, Ph.D. Thesis, Jilin University, 2003.
[2]	Eisenbud, D.; Green, M.; Harris, J., Cayley – bacharach theorems and conjectures, Bull. amer. math. soc. (N.S.), 33, 3, 295-324, (1996) · Zbl 0871.14024
[3]	Hartshorn, R., Algebraic geometry, (1977), Springer New York
[4]	Y.S. Lai, Some researches on piecewise algebraic curves and piecewise algebraic varieties, Ph.D. Thesis, DUT, 2002.
[5]	Shi, X.Q.; Wang, R.H., Bezout’s number for piecewise algebraic curves, Bit, 2, 339-349, (1999) · Zbl 0933.65017
[6]	Z.X. Su, Piecewise algebraic curves and surfaces and their applications in CAGD, Ph.D. Thesis, DUT, 1993.
[7]	Tan, S.L., Cayley – bacharach properly of an variety and Fujita’s conjecture, J. algebraic geom., 9, 201-222, (2000) · Zbl 0953.14033
[8]	Walker, R.J., Algebraic curves, (1950), Princeton University Press Princeton, NJ · Zbl 0039.37701
[9]	Wang, R.H., The structural characterization and interpolation for multivariate splines, Acta math. sinica, 18, 91-106, (1975), (English transl.ibid. 18(1975) 10-39) · Zbl 0358.41004
[10]	Wang, R.H., Multivariate spline and algebraic geometry, J. comput. appl. math., 121, 153-163, (2000) · Zbl 0960.41008
[11]	Wang, R.H., Multivariate spline functions and their applications, (2001), Science Press/Kluwer Pub Beijing/New York
[12]	Wang, R.H.; Lai, Y.S., Piecewise algebraic curve, J. comput. appl. math., 144, 277-289, (2002) · Zbl 1005.65021
[13]	Wang, R.H.; Xu, Z.Q., The estimation of Bezout’s number for piecewise algebraic curves, Sci. China ser A, 2, 185-192, (2003)
[14]	R.H. Wang, G.H. Zhao, An introduction to the piecewise algebraic curve, in: T. Mitsui (Ed.), Theory and Application of Scientific and Technical Computing, RIMS, Kyoto University, 1997, pp. 196-205. · Zbl 0942.65501
[15]	R.H. Wang, C.G. Zhu, Topology of piecewise algebraic curves, Math. Numer. Sinica 4 (2003), to appear.
[16]	R.H. Wang, C.G. Zhu, Nöther-type theorem of piecewise algebraic curves, Progr. Nat. Sci. 2 (2004), to appear.
[17]	Z.Q. Xu, Multivariate splines, piecewise algebraic curves and linear Diophantine equations, Ph.D. Thesis, DUT, 2003.