ON THE DIMENSION OF SPACES OF ALGEBRAIC CURVES PASSING THROUGH $ n $-INDEPENDENT NODES

Authors

  • H.A. Hakopyan Chair of Numerical Analysis and Mathematical Modeling, YSU, Armenia
  • H.M. Kloyan Chair of Numerical Analysis and Mathematical Modeling, YSU, Armenia

DOI:

https://doi.org/10.46991/PYSU:A/2019.53.2.091

Keywords:

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

Abstract

Let the set of nodes $ \LARGE{x} $ in the plain be $ n $-independent, i.e., each node has a fundamental polynomial of degree $ n $. Suppose also that $ \vert \LARGE{x} \normalsize \vert \mathclose{=} (n \mathclose{+} 1) \mathclose{+} n \mathclose{+} \cdots \mathclose{+} (n \mathclose{-} k \mathclose{+} 4) \mathclose{+} 2 $ and $ 3 \mathclose{\leq} k \mathclose{\leq} n \mathclose{-} 1 $. We prove that there can be at most 4 linearly independent curves of degree less than or equal to $ k $ passing through all the nodes of $ \LARGE{x} $. We provide a characterization of the case when there are exactly 4 such curves. Namely, we prove that then the set $ \LARGE{x} $ has a very special construction: all its nodes but two belong to a (maximal) curve of degree $ k \mathclose{-} 2 $. At the end, an important application to the Gasca-Maeztu conjecture is provided.

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Published

2019-08-15

Issue

Section

Mathematics

How to Cite

ON THE DIMENSION OF SPACES OF ALGEBRAIC CURVES PASSING THROUGH $ n $-INDEPENDENT NODES. (2019). Proceedings of the YSU A: Physical and Mathematical Sciences, 53(2 (249), 91-100. https://doi.org/10.46991/PYSU:A/2019.53.2.091