Extensions of planar GC sets and syzygy matrices
- PDF / 550,694 Bytes
- 19 Pages / 439.642 x 666.49 pts Page_size
- 13 Downloads / 214 Views
Extensions of planar GC sets and syzygy matrices ´ M. Carnicer1 Jesus
´ 2 · Carmen Godes
Received: 17 November 2017 / Accepted: 23 August 2018 / © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract The geometric characterization, introduced by Chung and Yao, identifies node sets for total degree interpolation such that the Lagrange fundamental polynomials are products of linear factors. Sets satisfying the geometric characterization are usually called GC sets. Gasca and Maeztu conjectured that planar GC sets of degree n contain n + 1 collinear points. It has been shown that the conjecture holds for degrees not greater than 5 but it is still unsolved for general degree. One promising approach consists of studying the syzygies of the ideal of polynomials vanishing at the nodes. In order to describe syzygy matrices of GC sets, we analyze the extension of a GC set of degree n to a GC set of degree n + 1, by adding a n + 2 nodes on a line. Keywords Total degree bivariate interpolation · Geometric characterization · Syzygy matrices Mathematics Subject Classification (2010) 41A05 · 41A63 · 13P10
1 Introduction The Lagrange interpolation problem in Pn , the space of polynomials of total degree not greater than n, consists of finding a polynomial p in Pn such that p(x) = f (x) for any x ∈ X, where X is a finite subset of Rk and f is a given function defined on a domain containing X. The points in the set X are usually called nodes.
Communicated by: Robert Schaback Jes´us M. Carnicer
[email protected] 1
Departamento de Matem´atica Aplicada/IUMA, Universidad de Zaragoza, Pedro Cerbuna 12, 50009 Zaragoza, Spain
2
Departamento de Matem´atica Aplicada, Universidad de Zaragoza, Carretera de Cuarte s/n, 22071 Huesca, Spain
J. M. Carnicer and C. God´es
Definition 1.1 A finite set of nodes X ⊂ Rk is Pn -correct if the Lagrange interpolation problem on X has a unique solution in Pn . Anecessary condition for a set of nodes X ⊂ Rk to be Pn -correct is that #X = n+k contrast to the univariate problem, not every subset of Rk , k > 1, consisting k . In n+k of k nodes is Pn -correct. Bivariate Pn -correct sets have remarkable geometric properties. For instance, no line in the plane can contain more than n + 1 nodes. Maximal lines, introduced by C. de Boor [2], are lines containing n + 1 nodes of a Pn -correct set. The properties of the maximal lines of a Pn -correct set lead to the Berzolari-Radon construction of Pn -correct sets [1, 19]. The number of maximal lines of a Pn -correct set, n ≥ 1, is at most n + 2. The defect of a Pn -correct set is the nonnegative integer computed as the difference between n + 2 and the number of maximal lines. In [15], Chung and Yao introduced a geometric characterization of Pn -correct sets such that their Lagrange fundamental polynomials are products of first-degree polynomials. Sets satisfying the geometric characterization for degree n are called GCn sets. Gasca and Maeztu [17] conjectured that, for each planar GCn set, there exists at least a maximal line. S
Data Loading...
