ON $n$-INDEPENDENT SETS LOCATED ON QUARTICS - Hakopian, H.A.; Malinyan, A.R. (Proceedings of the YSU A: Physical and Mathematical Sciences, 2013)
Denote the space of all bivariate polynomials of total degree $≤ n$ by $Π_n$. 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 ${\mathcal P}_{\mathcal X}:=\{p\in Π_n : p(x)=0, \forall x\in \mathcal X\}$ equals $\dim Π_n-#\mathcal X.$ Next, polynomial interpolation of degree $n$ is solvable only with these sets. Also the $n$-independent sets are exactly the subsets of $Π_n$-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 $δ,$ having the property $p\inΠ_n,~p(x)=0~\forall x\in {\mathcal X} \Rightarrow p=δ q,~q\in Π_{n-4}.$
Keywords: algebraic curve; ntal polynomial; $n$-independent point set; $n$-complete point set