  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.
