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

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.

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
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\textit{R. Wang} and \textit{C. Zhu}, J. Comput. Appl. Math. 163, No. 1, 269--276 (2004; Zbl 1070.14034)

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### References:

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