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 X are characterized by the fact that the dimension of the space P_X:={pΠ_n : p(x)=0, xX} equals Π_n-#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 X of quartic δ, having the property pΠ_n, p(x)=0 xX p=δ q, qΠ_n-4.