Bivariate Quasi-Tower Sets and Their Associated Lagrange Interpolation Bases

Details Bivariate Quasi-Tower Sets and Their Associated Lagrange Interpolation Bases Bivariate Quasi-Tower Sets and Their Associated Lagrange Interpolation Bases

2010-03-12

arxiv.org/abs/1003.2493v2

Mathematics

Commutative Algebra

17 pages, 2 figures, submitted to JCAM

Scientific paper

As we all known, there is still a long way for us to solve arbitrary multivariate Lagrange interpolation in theory. Nevertheless, it is well accepted that theories about Lagrange interpolation on special point sets should cast important lights on the general solution. In this paper, we propose a new type of bivariate point sets, quasi-tower sets, whose geometry is more natural than some known point sets such as cartesian sets and tower sets. For bivariate Lagrange interpolation on quasi-tower sets, we construct the associated degree reducing interpolation monomial and Newton bases w.r.t. common monomial orderings theoretically. Moreover, by inputting these bases into Buchberger-M\"{o}ller algorithm, we obtain the reduced Gr\"{o}bner bases for vanishing ideals of quasi-tower sets much more efficiently than before.

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