ON THE DIMENSION OF SPACES OF ALGEBRAIC CURVESPASSING THROUGH n-INDEPENDENT NODES

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ON THE DIMENSION OF SPACES OF ALGEBRAIC CURVESPASSING THROUGH n-INDEPENDENT NODES Hakopyan Hakopyan

Let the set of nodesXin the plain ben-independent, i.e., each node has afundamental polynomial of degreen.Suppose also that|X|= (n+1)+n+···++(n−k+4)+2 and 3≤k≤n−1.We prove that there can be at most 4 line-arly independent curves of degree less than or equal tokpassing through all thenodes ofX.We provide a characterization of the case when there are exactly 4such curves. Namely, we prove that then the setXhas a very special construc-tion: all its nodes but two belong to a (maximal) curve of degreek−2.At theend, an important application to the Gasca-Maeztu conjecture is provided.

DOI: 10.46991/PYSU:A/2019.53.2.091 Physical and Mathematical Sciences, 53(2 (249) 91-100

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Keywords

algebraic curves, n-independent nodes, maximal curves, Gasca-Maeztu conjecture.

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