The correspondence between multivariate spline ideals and piecewise algebraic varieties. (English summary) J. Comput. Appl. Math. 236 (2011), no. 5, 793–800. This paper establishes a connection between splines, i.e. piece-wise polynomial functions defined over bounded domains, in several variables and certain continuity conditions across the pieces ’ boundaries, and the zero sets that they define. Naturally, the concepts of spline ideals and piece-wise algebraic varieties come into play. Previous works on the algebro-geometric properties of spline spaces include [R. H. Wang, J. Comput. Appl. Math. 121 (2000), no. 1-2, 153–163; MR1780047; C. G. Zhu and R. H. Wang, J. Comput. Math. 23 (2005), no. 5, 503–512; MR2167179 (2006d:14071)]. In the current work an extension of the definition of radical ideals in the spline setting is presented. The main result is an analogue of Hilbert’s Nullstellensatz for spline ideals: The radical of a spline ideal is the vanishing ideal of the underlying variety
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