ON A RESULT CONCERNING ALGEBRAIC CURVES PASSING THROUGH п -INDEPENDENT NODES

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ON A RESULT CONCERNING ALGEBRAIC CURVES PASSING THROUGH п -INDEPENDENT NODES Hakop Hakopian

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.

DOI: 10.46991/PYSU:A/2022.56.3.097 Physical and Mathematical Sciences, 56(3 (259) 97-106

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Keywords

algebraic curve, maximal curve, fundamental polynomial, n-independent nodes

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