ON A RESULT CONCERNING ALGEBRAIC CURVES PASSING THROUGH $n$-INDEPENDENT NODES

Hakop A. Hakopian

doi:10.46991/PYSU:A/2022.56.3.097

Authors

Hakop A. Hakopian Chair of Numerical Analysis and Mathematical Modelling, YSU, Armenia

DOI:

https://doi.org/10.46991/PYSU:A/2022.56.3.097

Keywords:

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

Abstract

Let a set of nodes $\mathcal X$ in the plane be $n$-independent, i.e. each node has a fundamental polynomial of degree $n.$ Assume that $\#\mathcal X=d(n,n-3)+3= (n+1)+n+\cdots+5+3.$ 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 $\mathcal X.$ We provide a characterization of the case when there are exactly three such curves. Namely, we prove that then the set $\mathcal X$ 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.

References

Eisenbud D., Green M., Harris J. Cayley-Bacharach Theorems and Conjectures. Bull. Amer. Math. Soc. (N.S.) 33 (1996), 295-324.

Hakopian H., Jetter K., Zimmermann G. Vandermonde Matrices for Intersection Points of Curves. Jaen J. Approx. 1 (2009), 67-81.

Hakopian H., Malinyan A. Characterization of $n$-independent Sets with no More than $3n$ Points. Jaen J. Approx. 4 (2012), 121-136.

Hakopian H. On the Regularity of Multiariate Hermite Interpolation. Approx. Theory 105 (2000), 1-18. https://doi.org/10.1006/jath.1999.3345

Rafayelyan L. Poised Nodes Set Constructions on Algebraic Curves. East J. Approx. 17 (2011), 285-298.

Hakopian H., Toroyan S. On the Uniqueness of Algebraic Curves Passing Through $n$-independent Nodes. New York J. Math. 22 (2016), 441-452.

Walker R.J. Algebraic Curves. New Jersey, Princeton (1950).

Hakopian H., Kloyan H., Voskanyan D. On plane Algebraic Curves Passing Through $n$-independent Nodes. J. Cont. Math. Anal. 56 (2021), 280-294. https://doi.org/10.48550/arXiv.2105.13863

Hakopian H., Kloyan H. On the Dimension of Spaces of Algebraic Curves Passing Through $n$-independent nodes. Proceedings of the YSU. Phys. and Math. Sci. 53 (2019), 91-100. https://doi.org/10.46991/PYSU:A/2019.53.2.091