On the number of maximal planes of GC₂ sets in R³
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Gagik Ktryan [1]
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[1] Yerevan State University
Yerevan State University
Armenia
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- Localización: Jaen journal on approximation, ISSN 1889-3066, ISSN-e 1989-7251, Vol. 2, Nº. 1, 2010, págs. 129-143
- Idioma: inglés
- Texto completo no disponible (Saber más ...)
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Resumen
Sets of interpolation nodes satisfying the geometric condition (GC) of Chung and Yao are considered. A maximal hyperplane for a GCn set (a GC set of degree n) in Rd contains n+d−1 d−1 points of that set. There exist several conjectures about the number of maximal hyperplanes for a GC set. In the bivariate case, Gasca and Maeztu in 1982 conjectured that for every GCn set there exists at least a maximal line. This has been proved for n ≤ 4. Later on Carnicer and Gasca proved that for n ≤ 4, and for every GCn set, there exist at least 3 maximal lines and conjectured that this holds for n > 4. De Boor extended these conjectures to Rd: at least a maximal hyperplane as extension of the first one and at least d + 1 maximal hyperplanes as extension of the second one. The same author proved that the second conjecture does not hold for d > 2, showing a counterexample with d = 3, n = 2 and only 3 maximal hyperplanes. Recently, Apozyan et al. proved that for d = 3, n = 2, there exists at least one maximal hyperplane. In the present paper it is proved that, in fact, when d = 3, n = 2, there exist at least 3 maximal hyperplanes.
