 

Hakop Hakopian and Levon Rafayelyan
On a generalization of the GascaMaeztu conjecture view print


Published: 
May 27, 2015 
Keywords: 
GascaMaeztu conjecture, polynomial interpolation, algebraic curves, maximal line, maximal curve, poised, independent nodes. 
Subject: 
Primary: 41A05, 41A63; Secondary 14H50. 


Abstract
Denote the space of all
bivariate polynomials of total degree not exceeding n by Π_{n}.
The GascaMaeztu conjecture [Gasca M. and Maeztu J. I., On Lagrange
and Hermite interpolation in R^{k},
Numer. Math. 39 (1982), 114.] states that any Π_{n}poised set of nodes, all
fundamental polynomials of which are products of linear factors,
possesses a maximal line, i.e., a line passing through n+1 nodes.
Till now it is proved to be true for n≦ 5. The case n=5 was proved
recently in [Hakopian H., Jetter K. and Zimmermann G., The GascaMaeztu
conjecture for n=5, Numer. Math. 127 (2014), 685713]. In an earlier paper
the following generalized conjecture was proposed by the authors
of the present paper:
Any Π_{n}poised set of nodes, all fundamental polynomials of
which are reducible, possesses a maximal curve of some degree k,
1≦ k≦ n1, i.e., an algebraic curve passing through
(1/2)k(2nk+3) nodes. Clearly the two above conjectures coincide
in the case n≦ 2. In this paper we prove that the generalized
conjecture is true for n=3.


Author information
Hakop Hakopian:
Department of Informatics and Applied Mathematics, Yerevan State University, A. Manukyan Str. 1, 0025 Yerevan, Armenia
hakop@ysu.am
Levon Rafayelyan:
Department of Informatics and Applied Mathematics, Slavonic University, H. Emin Str. 123, 0051 Yerevan, Armenia
levon.rafayelyan@googlemail.com

