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Advances in Computational Mathematics

, Volume 45, Issue 2, pp 655–673 | Cite as

Extensions of planar GC sets and syzygy matrices

  • Jesús M. CarnicerEmail author
  • Carmen Godés
Article
  • 25 Downloads

Abstract

The geometric characterization, introduced by Chung and Yao, identifies node sets for total degree interpolation such that the Lagrange fundamental polynomials are products of linear factors. Sets satisfying the geometric characterization are usually called GC sets. Gasca and Maeztu conjectured that planar GC sets of degree n contain n + 1 collinear points. It has been shown that the conjecture holds for degrees not greater than 5 but it is still unsolved for general degree. One promising approach consists of studying the syzygies of the ideal of polynomials vanishing at the nodes. In order to describe syzygy matrices of GC sets, we analyze the extension of a GC set of degree n to a GC set of degree n + 1, by adding a n + 2 nodes on a line.

Keywords

Total degree bivariate interpolation Geometric characterization Syzygy matrices 

Mathematics Subject Classification (2010)

41A05 41A63 13P10 

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Notes

Funding information

Research partially supported by the Spanish Research Grant MTM2015-65433-P (MINECO/FEDER), by Gobierno the Aragón and Fondo Social Europeo.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada/IUMAUniversidad de ZaragozaZaragozaSpain
  2. 2.Departamento de Matemática AplicadaUniversidad de ZaragozaHuescaSpain

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