On bivariate fundamental polynomials

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.