New York Journal of Mathematics
Volume 21 (2015) 351-367

  

Hakop Hakopian and Levon Rafayelyan

On a generalization of the Gasca-Maeztu conjecture

view    print


Published: May 27, 2015
Keywords: Gasca-Maeztu 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 Πn. The Gasca-Maeztu conjecture [Gasca M. and Maeztu J. I., On Lagrange and Hermite interpolation in Rk, Numer. Math. 39 (1982), 1-14.] states that any Πn-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 Gasca-Maeztu conjecture for n=5, Numer. Math. 127 (2014), 685-713]. In an earlier paper the following generalized conjecture was proposed by the authors of the present paper: Any Πn-poised set of nodes, all fundamental polynomials of which are reducible, possesses a maximal curve of some degree k, 1≦ k≦ n-1, i.e., an algebraic curve passing through (1/2)k(2n-k+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