TY - JOUR
AU - Hakopian, H.A.
AU - Malinyan, A.R.
PY - 2013/04/10
Y2 - 2024/08/03
TI - ON $n$-INDEPENDENT SETS LOCATED ON QUARTICS
JF - Proceedings of the YSU A: Physical and Mathematical Sciences
JA - Proc. YSU A: Phys. Math. Sci.
VL - 47
IS - 1 (230)
SE - Mathematics
DO - 10.46991/PYSU:A/2013.47.1.006
UR - https://journals.ysu.am/index.php/proceedings-phys-math/article/view/vol47_no1_2013_pp006-012
SP - 6-12
AB - <p>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 $p\in\Pi_n,~p(x)=0~\forall x\in {\mathcal X} \Rightarrow p=\delta q,~q\in \Pi_{n-4}.$</p>
ER -