On bivariate fundamental polynomials
This provides theoretical bounds for the existence of factorized fundamental polynomials in bivariate polynomial interpolation, relevant to approximation theory.
The paper establishes that for bivariate interpolation, if the number of nodes is at most 2n+1, each node has a fundamental polynomial that is a product of lines; if at most 2n+⌊n/2⌋+1, each node has a fundamental polynomial that is a product of lines or conics. Counterexamples show these bounds are tight.
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.