Hakop Hakopian and Levon Rafayelyan
On a generalization of the Gasca-Maeztu conjecture
||May 27, 2015
||Gasca-Maeztu conjecture, polynomial interpolation, algebraic curves, maximal line, maximal curve, poised, independent nodes.
||Primary: 41A05, 41A63; Secondary 14H50.
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.
Department of Informatics and Applied Mathematics, Yerevan State University, A. Manukyan Str. 1, 0025 Yerevan, Armenia
Department of Informatics and Applied Mathematics, Slavonic University, H. Emin Str. 123, 0051 Yerevan, Armenia