ON A RESULT CONCERNING ALGEBRAIC CURVES PASSING THROUGH n-INDEPENDENT NODES - Hakopian, Hakop A. (Proceedings of the YSU A: Physical and Mathematical Sciences, 2022)
Let a set of nodes X in the plane be n-independent, i.e. each node has a fundamental polynomial of degree n. Assume that #X=d(n,n-3)+3= (n+1)+n+…+5+3. In this paper we prove that there are at most three linearly independent curves of degree less than or equal to n-1 that pass through all the nodes of X. We provide a characterization of the case when there are exactly three such curves. Namely, we prove that then the set X has a very special construction: either all its nodes belong to a curve of degree n-2, or all its nodes but three belong to a (maximal) curve of degree n-3. This result complements a result established recently by H. Kloyan, D. Voskanyan, and H. Hakopian. Note that the proofs of the two results are completely different.