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On a new property of n-poised and G C sets.
- Source :
-
Advances in Computational Mathematics . Jun2017, Vol. 43 Issue 3, p607-626. 20p. - Publication Year :
- 2017
-
Abstract
- In this paper we consider n-poised planar node sets, as well as more special ones, called G C 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 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 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 sets. At the end we present a conjecture concerning any k-node line. [ABSTRACT FROM AUTHOR]
- Subjects :
- *SET theory
*POLYNOMIALS
*UNIVARIATE analysis
Subjects
Details
- Language :
- English
- ISSN :
- 10197168
- Volume :
- 43
- Issue :
- 3
- Database :
- Academic Search Index
- Journal :
- Advances in Computational Mathematics
- Publication Type :
- Academic Journal
- Accession number :
- 123057975
- Full Text :
- https://doi.org/10.1007/s10444-016-9499-3