Algebra-geometry of piecewise algebraic varieties
A piecewise algebraic variety is the zero-locus of a collection of splines -- piecewise polynomials of given differentiability class on a fixed partition of affine space. Building on earlier results by the same authors, this article develops the theory of spline ideals and their relation to piecewise algebraic varieties. Some fundamental results, like the Hilbert's Basis Theorem, carry over from the purely polynomial theory, others, like Hilbert's Nullstellensatz, don't. Interestingly enough, the failure of the Nullstellensatz can be related to the topology and even the geometry of the underlying partition. In some special cases, Hilbert's Nullstellensatz holds true.
scientific article; zbMATH DE number 6121160