Geodesic
On a conjecture in bivariate interpolation
Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 1 (2016), pp. 30-34 Cet article a éte moissonné depuis la source Math-Net.Ru

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Denote the space of all bivariate polynomials of total degree $\leq n$ by $\Pi_n$. We are interested in $n$-poised sets of nodes with the property that the fundamental polynomial of each node is a product of linear factors. In 1981 M. Gasca and J.I.Maeztu conjectured that every such set contains necessarily $n+1$ collinear nodes. Up to now this had been confirmed for degrees $n\leq5$. Here we bring a simple and short proof of the conjecture for $n=4$.
Keywords: poised, independent nodes, algebraic curves.
Mots-clés : polynomial interpolation 
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S. Z. Toroyan. On a conjecture in bivariate interpolation. Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 1 (2016), pp. 30-34. http://geodesic.mathdoc.fr/item/UZERU_2016_1_a4/

[1] D. Eisenbud, M. Green, J. Harris, “Cayley–Bacharach Theorems and Conjectures”, Bull. Amer. Math. Soc., 33 (1996), 295–324 | DOI | MR | Zbl

[2] H. Hakopian, K. Jetter, G. Zimmermann, “Vandermonde Matrices for Intersection Points of Curves”, Jaen J. Approx., 1 (2009), 67–81 | MR | Zbl

[3] M. Gasca, J. I. Maeztu, “On Lagrange and Hermite Interpolation in $\mathbb{R}^k $”, Numer. Math., 39 (1982), 1–14 | DOI | MR | Zbl

[4] H. Hakopian, K. Jetter, G. Zimmermann, “The Gasca–Maeztu Conjecture for $n = 5$”, Numer. Math., 127 (2014), 685–713 | DOI | MR | Zbl

[5] V. Bayramyan, H. Hakopian, S. Toroyan, “A Simple Proof of the Gasca–Maeztu Conjecture for $n=4$”, Jáen J. Approx. Theory, 7 (2015), 137–147

[6] J.R. Busch, “A Note on Lagrange Interpolation in $\mathbb{R}^2$”, Rev. Un. Mat. Argentina, 36 (1990), 33–38 | MR | Zbl