On $n$-independent sets located on quartics

Denote the space of all bivariate polynomials of total degree $\leq n$ by $\Pi_n$. We study the $n$-independence of points sets on quartics, i.e. on algebraic curves of degree $4$. The $n$-independent sets $\mathcal X$ are characterized by the fact that the dimension of the space $\mathcal P_{\mathcal X}:=\{p\in \Pi_n : p(x) =0,\forall x \in\mathcal X\}$ equals $\hbox{dim}\Pi_n-\#\mathcal X$. Next, polynomial interpolation of degree $n$ is solvable only with these sets. Also the $n$-independent sets are exactly the subsets of $\Pi_n$-poised sets. In this paper we characterize all $n$-independent sets on quartics. We also characterize the set of points that are $n$-complete in quartics, i.e. the subsets $\mathcal X$ of quartic $\delta$, having the property $p\in\Pi_n, p(x)=0 \ \forall x \in \mathcal X \to p=\delta q, q \in \Pi_{n-4}$.