Abstract
An \(n\)-poised set in two dimensions is a set of nodes admitting unique bivariate interpolation with polynomials of total degree at most \(n\). We are interested in poised sets with the property that all fundamental polynomials are products of linear factors. Gasca and Maeztu (Numer Math 39:1–14, 1982) conjectured that every such set necessarily contains \(n+1\) collinear nodes. Up to now, this had been confirmed only for \(n\le 4\), the case \(n=4\) having been proved for the first time by Busch (Rev Un Mat Argent 36:33–38, 1990). In the present paper, we prove the case \(n=5\) with new methods that might also be useful in deciding the still open cases for \(n\ge 6\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berzolari, L.: Sulla determinazione d’una curva o d’una superficie algebrica e su alcune questioni di postulazione. Ist. Lomb. Rend. (II. Ser.) 47, 556–564 (1914)
de Boor, C.: Multivariate polynomial interpolation: conjectures concerning GC-sets. Numer. Algorithms 45, 113–125 (2007)
Busch, J.R.: A note on Lagrange interpolation in \({\mathbb{R}}^2\). Rev. Un. Mat. Argent. 36, 33–38 (1990)
Carnicer, J.M., Gasca, M.: Planar configurations with simple Lagrange interpolation formulae. In: Lyche, T., Schumaker, L.L. (eds.) Mathematical Methods in Curves and Surfaces: Oslo 2000, pp. 55–62. Vanderbilt University Press, Nashville (2001)
Carnicer, J.M., Gasca, M.: A conjecture on multivariate polynomial interpolation. Rev. R. Acad. Cienc. Exactas Fís. Nat. (Esp.). Ser. A Mat 95, 145–153 (2001)
Carnicer, J.M., Gasca, M.: On Chung and Yao’s geometric characterization for bivariate polynomial interpolation. In: Lyche, T., Mazure, M.-L., Schumaker, L.L. (eds.) Curve and Surface Design: Saint Malo 2002, 21–30. Nashboro Press, Brentwood (2003)
Chung, K.C., Yao, T.H.: On lattices admitting unique Lagrange interpolations. SIAM J. Numer. Anal. 14, 735–743 (1977)
Gasca, M., Maeztu, J.I.: On Lagrange and Hermite interpolation in \({\mathbb{R}}^k\). Numer. Math. 39, 1–14 (1982)
Hakopian, H.: On a class of Hermite interpolation problems. Adv. Comput. Math. 12, 303–309 (2000)
Hakopian, H., Jetter, K., Zimmermann, G.: Vandermonde matrices for intersection points of curves. Jaén J. Approx. 1, 67–81 (2009)
Hakopian, H., Jetter, K., Zimmermann, G.: A new proof of the Gasca-Maeztu conjecture for \(n=4\). J. Approx. Theory 159, 224–242 (2009)
Hakopian, H., Malinyan, A.: Characterization of \(n\)-independent sets with no more than \(3n\) points. Jaén J. Approx. 4, 121–136 (2012)
Hakopian, H., Rafayelyan, L.: On a generalization of Gasca-Maeztu conjecture (preprint)
Jetter, K.: Some contributions to bivariate interpolation and cubature. In: Chui, C.K., Schumaker, L.L., Ward, J.D. (eds.) Approximation Theory IV, pp. 533–538. New York, Academic Press (1983)
Ktryan, G.: On the number of maximal planes of \(GC_2\) sets in \({\mathbb{R}}^3\). Jaén J. Approx. 2, 129–143 (2010)
Radon, J.: Zur mechanischen Kubatur. Monatsh. Math. 52, 286–300 (1948)
Rafayelyan, L.: Poised nodes set constructions on algebraic curves. East J. Approx. 17, 285–298 (2011)
Acknowledgments
We would like to thank the two referees for their very thorough work and their many constructive comments, some of which might well be useful in the further investigation of the Gasca–Maeztu conjecture. Furthermore, we gratefully acknowledge support from the DAAD, grant # A/12/05111, and from the State Committee of Science of Armenia, grant # 11-1A290, which enabled the first author to visit the other two for an extended time.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Hakopian, H., Jetter, K. & Zimmermann, G. The Gasca–Maeztu conjecture for \(n=5\) . Numer. Math. 127, 685–713 (2014). https://doi.org/10.1007/s00211-013-0599-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-013-0599-4