2015/10/18 by Hakop Hakopian, Hakopian, H., S. Toroyan +1 · 2 citations
Mathematics · #14H50 #41A05 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.1510.05211
openalex publication_date 2015/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A set of nodes is called n-independent if each its node has a fundamental polynomial of degree n. We proved in a previous paper [H. Hakopian and S. Toroyan, On the minimal number of nodes determining uniquelly algebraic curves, accepted in Proceedings of YSU] that the minimal number of n-independent nodes determining uniquely the curve of degree k≤ n equals to \mathcal K:=(1/2)(k-1)(2n+4-k)+2. Or, more precisely, for any n-independent set of cardinality \mathcal K there is at most one curve of degree k≤ n passing through its nodes, while there are n-independent node sets of cardinality \mathcal K-1 through which pass at least two such curves. In this paper we bring a simple characterization of the latter sets. Namely, we prove that if two curves of degree k≤ n pass through the nodes of an n-independent node set \mathcal X of cardinality \mathcal K-1 then all the nodes of \mathcal X but one belong to a (maximal) curve of degree k-1.