Extensions of planar GC sets and syzygy matrices

Carnicer, J. M.; Godés, Carmen

doi:10.1007/s10444-018-9630-8

Cited by 3 publications

(2 citation statements)

References 13 publications

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“…In the sequel we will need the following characterization of GC n set X , with #M (X ) = n − 1, due to Carnicer and Godés (see [10], Section 5, Case d=3, for a proof detail): Let us denote by dd 1 , dd 2 , dd 3 , the lines passing through the latter triples, respectively, and call them DD-lines. Also, the nodes D i are called D-nodes.…”

Section: Defect 3 Setsmentioning

confidence: 99%

On the basic properties of $GC_n$ sets

Hakopian¹,

Vardanyan²

2020

Preprint

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A planar node set X , with #X = n+2 2 , is called GC n set if each node possesses fundamental polynomial in form of a product of n linear factors. We say that a node uses a line if the line is a factor of the fundamental polynomial of the node. A line is called k-node line if it passes through exactly k-nodes of X . At most n + 1 nodes can be collinear in any GC n set and an (n + 1)-node line is called a maximal line. The Gasca-Maeztu conjecture (1982) states that every GC n set has a maximal line. Until now the conjecture has been proved only for the cases n ≤ 5.Here, for a line we introduce and study the concept of -lowering of the set X and define so called proper lines. We also provide refinements of several basic properties of GC n sets regarding the maximal lines, nnode lines, the used lines, as well as the subset of nodes that use a given line.

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Section: Defect 3 Setsmentioning

confidence: 99%

On the basic properties of $GC_n$ sets

Hakopian¹,

Vardanyan²

2020

Preprint

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“…A promising approach, based on methods from algebraic geometry, was introduced in [12] and suggests to consider so-called syzygy matrices that capture information about the structure of the underlying interpolation problem. In this paper, which is a continuation of [8,9], we collect further information about these syzygy matrices and explicitly show their behavior in certain cases of a given defect. Even if this is not yet a proof of the conjecture we consider these results useful for further studying this connection between interpolation problems and the associated syzygy matrices.…”

Section: Introductionmentioning

confidence: 99%

Radon’s construction and matrix relations generating syzygies

2020

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Let Π n be the set of bivariate polynomials of degree not greater than n. A Π n -correct set of nodes is a set such that the Lagrange interpolation problem with respect to these nodes has a unique solution. A maximal line of a Π n -correct set is any line containing exactly n+1 nodes. Syzygy matrices can be used to find linear factors of the fundamental polynomials and detect maximal lines. We suggest to use matrix relations in order to generate syzygies, identify linear factors of fundamental polynomials and detect maximal lines. We interpret our results in the important case of GC sets trying to shed some light on the Gasca-Maeztu conjecture.

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On the usage of lines in GCn sets

Hakopian

Vardanyan

2019

Adv Comput Math

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A planar node set X , with |X | = n+2 2 is called GC n set if each node possesses fundamental polynomial in form of a product of n linear factors. We say that a node uses a line Ax+By +C = 0 if Ax+By +C divides the fundamental polynomial of the node. A line is called k-node line if it passes through exactly k-nodes of X . At most n + 1 nodes can be collinear in GC n sets and an (n + 1)-node line is called maximal line. The Gasca -Maeztu conjecture (1982) states that every GC n set has a maximal line. Until now the conjecture has been proved only for the cases n ≤ 5. Here we adjust and prove a conjecture proposed in the paper -V. Bayramyan, H. H., Adv Comput Math, 43: 607-626, 2017. Namely, by assuming that the Gasca-Maeztu conjecture is true, we prove that for any GC n set X and any k-node line the following statement holds: Either the line is not used at all, or it is used by exactly s 2 nodes of X , where s satisfies the condition σ := 2k − n − 1 ≤ s ≤ k. If in addition σ ≥ 3 and µ(X ) > 3 then the first case here is excluded, i.e., the line is necessarily a used line. Here µ(X ) denotes the number of maximal lines of X .At the end, we bring a characterization for the usage of k-node lines in GC n sets when σ = 2 and µ(X ) > 3.

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Extensions of planar GC sets and syzygy matrices

Cited by 3 publications

References 13 publications

On the basic properties of $GC_n$ sets

On the basic properties of $GC_n$ sets

Radon’s construction and matrix relations generating syzygies

On the usage of lines in GCn sets

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