Geometric characterization and generalized principal lattices

Abstract

The sets of nodes in the plane for which its nth degree Lagrange polynomials can be factored as a product of first degree polynomials satisfy a geometric characterization: for each node there exists a set of n lines containing the other nodes. Generalized principal lattices are sets of nodes defined by three families of lines. A generalized principal lattice satisfies the geometric characterization and there exist exactly three lines in the plane containing more nodes than the degree. In this paper, we show a converse, valid for degrees n 7: if a set of nodes satisfy the geometric characterization and there exist exactly three lines containing n + 1 nodes, then it is a generalized principal lattice.

Classification of sets satisfying the geometric characterization

“…In Theorem 3.6 of [10], it was proved that all GC n,n−1 sets with n ≤ ν + 3 are GPL n sets. Taking into account that ν ≥ 4, we have that any GC n,n−1 set with n ≤ 7 is a GPL n set.…”

Generalized principal lattices and cubic pencils

“…A more general definition was provided in [10], which we shall use throughout this paper. For a multivariate definition see [9].…”

“…By Proposition 2.5 (c) of [10], the set X determines the lines L 0 0:2 (X), L 1 0:2 (X), L 2 0:2 (X). Let us consider the point y 022 := L 1 2 ∩ L 2 2 and L any arbitrary line that passing through y 022 .…”

“…In [7,8], principal lattices were generalized by permitting that the three families of lines involved in the definition of principal lattices not to be necessarily linear pencils. These sets of nodes, called GPL n sets, have good properties for interpolation because they also satisfy the geometric characterization of Chung and Yao. Recently, the original definition of planar GPL n sets has been further generalized in [10]. This change in the definition has been necessary for showing that GPL n sets coincide with GC n sets of defect n − 1 (see [10]).…”

“…These sets of nodes, called GPL n sets, have good properties for interpolation because they also satisfy the geometric characterization of Chung and Yao. Recently, the original definition of planar GPL n sets has been further generalized in [10]. This change in the definition has been necessary for showing that GPL n sets coincide with GC n sets of defect n − 1 (see [10]). The new definition of GPL n sets can be found in Definition 2.1 and represents a wider class of GC n sets.…”