NDLI: On a new property of n-poised and G C n sets

Content Provider	Springer Nature Link
Author	Hakopian, Hakop Bayramyan, Vahagn
Copyright Year	2016
Abstract	In this paper we consider n-poised planar node sets, as well as more special ones, called G C n sets. For the latter sets each n-fundamental polynomial is a product of n linear factors as it always holds in the univariate case. A line ℓ is called k-node line for a node set $\mathcal X$ if it passes through exactly k nodes. An (n + 1)-node line is called maximal line. In 1982 M. Gasca and J. I. Maeztu conjectured that every G C n set possesses necessarily a maximal line. Till now the conjecture is confirmed to be true for n ≤ 5. It is well-known that any maximal line M of $\mathcal X$ is used by each node in $\mathcal X\setminus M, $ meaning that it is a factor of the fundamental polynomial. In this paper we prove, in particular, that if the Gasca-Maeztu conjecture is true then any n-node line of G C n set $\mathcal {X}$ is used either by exactly $\binom {n}{2}$ nodes or by exactly $\binom {n-1}{2}$ nodes. We prove also similar statements concerning n-node or (n − 1)-node lines in more general n-poised sets. This is a new phenomenon in n-poised and G C n sets. At the end we present a conjecture concerning any k-node line.
Ending Page	626
Page Count	20
Starting Page	607
File Format	PDF
ISSN	10197168
e-ISSN	15729044
Journal	Advances in Computational Mathematics
Issue Number	3
Volume Number	43
Language	English
Publisher	Springer US
Publisher Date	2016-11-16
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	n-poised set Fundamental polynomial G C n set Visualization Computational Mathematics and Numerical Analysis Gasca-Maeztu conjecture Maximal curve Plane and space curves Mathematical and Computational Biology Computational Science and Engineering n-independent set Multidimensional problems (should also be assigned at least one other classification number in this section) Interpolation Maximal line Polynomial interpolation Mathematical Modeling and Industrial Mathematics Algebraic curve
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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