Bivariate Quasi-Tower Sets and Their Associated Lagrange Interpolation Bases
For researchers in multivariate polynomial interpolation, this work provides a new point set and efficient basis construction, though it is incremental as it extends existing tower set concepts.
This paper introduces bivariate quasi-tower sets for Lagrange interpolation, which have a more natural geometry than cartesian or tower sets. The authors construct degree-reducing interpolation bases and show that using these bases in the Buchberger-Möller algorithm yields reduced Gröbner bases more efficiently than previous methods.
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-Möller algorithm, we obtain the reduced Gröbner bases for vanishing ideals of quasi-tower sets much more efficiently than before.