International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 473582, 12 pages
Research Article

An Upper Bound of the Bezout Number for Piecewise Algebraic Curves over a Rectangular Partition

School of Mathematical Sciences, Ocean University of China, Qingdao, Shandong 266100, China

Received 24 March 2012; Accepted 10 June 2012

Academic Editor: Raül Curto

Copyright © 2012 Feng-Gong Lang and Xiao-Ping Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A piecewise algebraic curve is a curve defined by the zero set of a bivariate spline function. Given two bivariate spline spaces 𝑆 𝑟 𝑚 (Δ) and 𝑆 𝑡 𝑛 (Δ) over a domain D with a partition Δ, the Bezout number BN(m,r;n,t;Δ) is defined as the maximum finite number of the common intersection points of two arbitrary piecewise algebraic curves 𝑓 ( 𝑥 , 𝑦 ) = 0 and 𝑔 ( 𝑥 , 𝑦 ) = 0 , where 𝑓 ( 𝑥 , 𝑦 ) 𝑆 𝑟 𝑚 (Δ) and 𝑔 ( 𝑥 , 𝑦 ) 𝑆 𝑡 𝑛 (Δ). In this paper, an upper bound of the Bezout number for piecewise algebraic curves over a rectangular partition is obtained.