Bivariate Quasi-Tower Sets and Their Associated Lagrange Interpolation Bases

TL;DR

This paper introduces quasi-tower sets, a new class of bivariate point sets, and develops associated interpolation bases and efficient algorithms for their algebraic properties, advancing multivariate Lagrange interpolation theory.

Contribution

It proposes quasi-tower sets with natural geometry and constructs associated bases, improving the efficiency of computing Gröbner bases for these sets.

Findings

Constructed degree reducing interpolation monomial bases

Developed Newton bases for quasi-tower sets

Enhanced efficiency of Gröbner basis computation

Abstract

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.