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Extensions of planar GC sets and syzygy matrices

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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.

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Funding

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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Authors and Affiliations

  1. Departamento de Matemática Aplicada/IUMA, Universidad de Zaragoza, Pedro Cerbuna 12, 50009, Zaragoza, Spain

    Jesús M. Carnicer

  2. Departamento de Matemática Aplicada, Universidad de Zaragoza, Carretera de Cuarte s/n, 22071, Huesca, Spain

    Carmen Godés

Authors
  1. Jesús M. Carnicer
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  2. Carmen Godés
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Correspondence to Jesús M. Carnicer.

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Communicated by: Robert Schaback

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Carnicer, J.M., Godés, C. Extensions of planar GC sets and syzygy matrices. Adv Comput Math 45, 655–673 (2019). https://doi.org/10.1007/s10444-018-9630-8

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  • DOI: https://doi.org/10.1007/s10444-018-9630-8

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