Algebra-geometry of piecewise algebraic varieties
type: Item
instance of: scholarly article
title: Algebra-geometry of piecewise algebraic varieties
zbMATH Open document ID: 1257.14038
author: Chun-Gang Zhu
author: Ren-Hong Wang
publication date: 2013-01-02T00:00:00Z
review text: 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.
reviewed by: Hans-Peter Schröcker
Mathematics Subject Classification ID: 14Q99
Mathematics Subject Classification ID: 41A15
Mathematics Subject Classification ID: 65D07
Mathematics Subject Classification ID: 65D18
zbMATH DE Number: 6121160
zbMATH Keywords: piecewise algebraic variety
zbMATH Keywords: multivariate spline
zbMATH Keywords: partition
zbMATH Keywords: algebraic geometry
MaRDI profile type: Publication
full work available at URL: s10114-012-9552-1
OpenAlex ID: W2032847817
cites work: A practical guide to splines
cites work: Q5841010
cites work: Q4130491
cites work: Multivariate spline and algebraic geometry
cites work: Piecewise algebraic varieties
cites work: Nöther-type theorem of piecewise algebraic curves
cites work: The Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations
cites work: Q5699649
cites work: Lagrange interpolation by bivariate splines on cross-cut partitions
cites work: Nöther-type theorem of piecewise algebraic curves on triangulation
cites work: Nöther-type theorem of piecewise algebraic curves on quasi-cross-cut partition
DOI: 10.1007/S10114-012-9552-1
published in: Acta Mathematica Sinica, English Series
description: scientific article; zbMATH DE number 6121160