ON n-INDEPENDENT SETS LOCATED ON QUARTICS

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Submitted: 2025-02-18; Published: 2025-02-18

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Abstract

Denote the space of all bivariate polynomials of total degree≤nbyΠn. We studythen-independence of points sets on quartics, i.e. on algebraic curves of degree 4.Then-independent setsXare characterized by the fact that the dimension of the spacePX:={p∈Πn:p(x) =0,∀x∈X}equals dimΠn−#X.Next, polynomial interpolationof degreenis solvable only with these sets. Also then-independent sets are exactly thesubsets ofΠn-poised sets. In this paper we characterize alln-independent sets on quartics.We also characterize the set of points that aren-complete in quartics, i.e. the subsetsXofquarticδ,having the propertyp∈Πn,p(x) =0∀x∈X⇒p=δq,q∈Πn−4.