On bivariate fundamental polynomials

TL;DR

This paper investigates conditions under which nodes in bivariate polynomial interpolation have fundamental polynomials composed of lines or conics, establishing bounds on the number of nodes for these properties to hold.

Contribution

It proves that nodes have fundamental polynomials as products of lines or conics within certain node count bounds, and provides counterexamples beyond these bounds.

Findings

Fundamental polynomials are products of lines if node count ≤ 2n+1.

Fundamental polynomials are products of lines or conics if node count ≤ 2n+[n/2]+1.

Counterexamples show the bounds are sharp and do not hold beyond these node counts.

Abstract

An $n$-independent set in two dimensions is a set of nodes admitting (not necessarily unique) bivariate interpolation with polynomials of total degree at most $n.$ For an arbitrary $n$-independent node set $\mathcal X$ we are interested with the property that each node possesses an $n$-fundamental polynomial in form of product of linear or quadratic factors. In the present paper we show that each node of $\mathcal X$ has an $n$-fundamental polynomial, which is a product of lines, if $\#\mathcal X\le 2n+1.$ Next we prove that each node of $\mathcal X$ has an $n$-fundamental polynomial, which is a product of lines or conics, if $\#\mathcal X\le 2n+[n/2]+1$. We have a counterexample in each case to show that the results are not valid in general if $\#\mathcal X\ge 2n+2$ and $\#\mathcal X\ge 2n+[n/2]+2,$ respectively.