ON $n$-INDEPENDENT SETS LOCATED ON QUARTICS

Abstract

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 $\dim \Pi_n-\#\mathcal X.$ 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 $p\in\Pi_n,~p(x)=0~\forall  x\in {\mathcal X} \Rightarrow p=\delta q,~q\in \Pi_{n-4}.$